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前言
FOREWORD
今天，在《头脑风暴》出版四十年后重读它 ，我有两种相互矛盾的反应：
IN REREADING MINDSTORMS TODAY, FORTY YEARS AFTER ITS PUBLICATION, I had two conflicting reactions:
一方面，西摩的许多思想在 1980 年被视为激进，如今已成为教育主流的一部分。
On the one hand, so many of Seymour’s ideas that were seen as radical in 1980 are now part of the education mainstream.
另一方面，西摩的许多梦想仍未实现。
On the other hand, so many of Seymour’s dreams remain unrealized and unfulfilled.
为什么西摩的理念与当今现实如此契合，但又如此脱节？
How is it that Seymour’s ideas can be so aligned with today’s realities but still so disconnected from them?
为了理解这种看似矛盾的情况，让我们回顾一下 1980 年，即《头脑风暴》出版的那段时间。当时第一台个人电脑才刚刚问世几年。当时还没有人拥有手机、平板电脑甚至笔记本电脑。当时还没有网络，很少有人听说过互联网。因此，像西摩当时那样预测全世界数以百万计的儿童很快就会像现在一样每天接触数字技术，这确实是一件非常激进的事情。
To make sense of this seeming contradiction, it’s helpful to transport yourself back to 1980, when Mindstorms was published. The first personal computers had been developed just a few years earlier. No one had mobile phones or tablets or even laptops. The Web didn’t exist, and few people had heard of the Internet. So it was truly radical to predict, as Seymour did then, that millions and millions of children around the world would soon be interacting with digital technologies every day, as they do now.
西摩对儿童使用电脑的设想甚至更为激进。1980 年，一小群研究人员开始思考将电脑用于K -12 教育，其中大多数人专注于“计算机辅助教学”，即让电脑扮演传统教师的角色：向学生传递信息和指导，进行测验以衡量学生的学习成果，然后根据学生的反应调整后续教学。
Even more radical were the ways in which Seymour imagined children using computers. In the small community of researchers who in 1980 were beginning to think about the use of computers in k–12 education, most focused on “computer-aided instruction,” in which computers played the role of a traditional teacher: delivering information and instruction to students, conducting quizzes to measure what the students had learned, then adapting subsequent instruction based on student responses.
在《头脑风暴》中，西摩提出了截然不同的观点。对于西摩来说，计算机并不是教师的替代品，而是一种孩子们可以用来制作东西和表达自己的新媒介。在《头脑风暴》中，西摩用了一个特别令人难忘的措辞，拒绝了“用计算机来给孩子编程”的计算机辅助教学方法，并主张一种“孩子给计算机编程”的替代方法。
In Mindstorms, Seymour offered a radically different vision. For Seymour, computers were not a replacement for the teacher but a new medium that children could use for making things and expressing themselves. Using a particularly memorable turn of phrase in Mindstorms, Seymour rejected the computer-aided instruction approach in which “the computer is being used to program the child” and argued for an alternative approach in which “the child programs the computer.”
自《头脑风暴》出版以来的几十年里，西摩关于教育技术的理念影响力日渐增强。如今，世界各地的学校都增设了创客空间和编程课程，为学生提供了 1980 年很少有人能想象到的机会。西摩的作品应被视为创客运动和编程运动的思想灵感。
Seymour’s ideas about educational technologies have had a growing influence in the decades since the publication of Mindstorms. Schools everywhere are now adding makerspaces and coding classes, offering students opportunities that few could have imagined in 1980. Seymour’s work should be seen as the intellectual inspiration for the Maker Movement and the Coding Movement.
然而，如果 Seymour 还活着，我毫不怀疑他会对学校引入制作和编码的方式感到非常沮丧。Seymour 认为，目前大多数举措都是“以技术为中心”（Seymour 推广的术语）。也就是说，这些举措过于注重帮助孩子们发展技术技能：如何使用 3D打印机、如何定义算法、如何编写高效的计算机代码。
Yet if Seymour were alive today, I have no doubt that he would be very frustrated with the ways that making and coding are being introduced in schools. Seymour would view most of the current initiatives as “technocentric” (a term that Seymour popularized). That is, the initiatives focus too much on helping children develop technical skills: how to use a 3d printer, how to define an algorithm, how to write efficient computer code.
对于西摩来说，技术技能从来都不是目标。在《头脑风暴》的导言中，他写道：“我关注的重点不是机器，而是思维。”西摩当然对机器和新技术感兴趣，但仅限于它们能够支持学习或带来关于学习的新见解。
For Seymour, technical skills were never the goal. In the Introduction to Mindstorms, he wrote: “My central focus is not on the machine but on the mind.” Seymour was certainly interested in machines and new technologies, but only insofar as they could support learning or lead to new insights about learning.
《头脑风暴》的很大一部分内容都集中在 logo 上，这是第一种专门为儿童设计的编程语言。但《头脑风暴》的核心是西摩关于教育和学习的理念，而不是技术问题。在书中，他为后来被他命名为“建构主义”的教育理论奠定了思想基础。该理论建立在伟大的儿童发展先驱让·皮亚杰的工作基础之上，西摩曾在 20 世纪 60 年代初与皮亚杰合作过。皮亚杰的伟大见解是，知识不是从老师传给学习者的；相反，儿童通过与周围的人和事物的日常互动不断建构知识。西摩的建构主义理论增加了第二种建构类型，认为当儿童积极参与构建世界上的事物时，他们构建知识的效率最高。当儿童构建世界上的事物时，他们会在头脑中构建新的想法和理论，从而激励他们在世界上构建新事物，如此反复。
A significant portion of Mindstorms focuses on logo, the first programming language designed specifically for children. But at the core of Mindstorms are Seymour’s ideas about education and learning, not technical issues. In the book, he lays the intellectual foundation for an educational theory that he later named “constructionism.” The theory builds on the work of Jean Piaget, the great child-development pioneer, who Seymour had collaborated with in the early 1960s. Piaget’s great insight was that knowledge is not delivered from teacher to learner; rather, children are constantly constructing knowledge through their everyday interactions with people and objects around them. Seymour’s constructionism theory adds a second type of construction, arguing that children construct knowledge most effectively when they are actively engaged in constructing things in the world. As children construct things in the world, they construct new ideas and theories in their minds, which motivates them to construct new things in the world, and on and on.
西摩在各种“建造”活动中都看到了丰富的学习机会：在沙滩上建造沙堡、在日记中写故事、在速写本上画画。西摩为什么对计算机技术如此感兴趣？因为他认识到计算机技术可以极大地扩展儿童创造的范围和方式。有了计算机，孩子们可以创造移动、交互和随时间变化的东西，如动画、模拟和互动游戏。在这个过程中，孩子们可以对周围世界动态系统的运作方式获得新的见解——包括他们自己思维的运作方式。此外，计算机使孩子们能够以前所未有的方式修改、复制、记录和分享他们的作品，为他们探索和理解创作过程提供了新的方式。
Seymour saw rich learning opportunities in all different types of “construction” activities: building sand castles on the beach, writing stories in a diary, drawing pictures in a sketchbook. Why was Seymour so interested in computational technologies? Because he recognized that computational technologies can greatly expand the range of what and how children create. With computers, children can create things that move, interact, and change over time, such as animations, simulations, and interactive games. In the process, children can gain new insights into the workings of dynamic systems in the world around them—including the workings of their own minds. In addition, computers enable children to modify, duplicate, document, and share their creations in ways they never could before, providing new ways for them to explore and understand the creative process.
人们似乎只听取了他的部分信息，往往只关注技术而忽略了理念，这一直让 Seymour 感到沮丧。在《头脑风暴》出版 20 年后，Seymour 在一篇文章中感叹，这本书的副标题中的三个部分——儿童、计算机和强大的理念——并没有得到同等的重视：“许多从这本书中获得启发和肯定的教育工作者（以及那些讨厌这本书的人）大部分在讨论这本书时，都把它当成是关于儿童和计算机的，好像第三个术语只是一个简短的片段，是教育技术讨论中普遍存在的一种口号。我并不是这个意思：我真的以为自己在写一本关于理念的书！”
It continually frustrated Seymour that people seemed to hear only part of his message, often focusing on the technology at the expense of the ideas. In an article Seymour wrote twenty years after the publication of Mindstorms, he lamented that the three parts of the book’s subtitle—Children, Computers, and Powerful Ideas—had not been equally appreciated: “Most of the many educators who found inspiration and affirmation in the book (as well as those who hated it) discussed it as if it were about children and computers, as if the third term was there as a sound bite, the kind of shibboleth that pervades the discourse of technology in education. I did not mean it to be that: I actually thought I was writing a book about ideas!”
当然，当今学习编码计划的支持者会说，他们感兴趣的不仅仅是技术技能，还有理念。他们中的许多人将自己的工作框架设在“计算思维”的理念上——旨在向儿童介绍来自计算机科学领域但适用于许多其他领域的解决问题的策略。学习解决问题的策略当然很有价值。但西摩有更大、更广阔的视野。他希望不仅帮助孩子们发展思维，还帮助他们发展自己的声音。
Of course, proponents of today’s learn-to-code initiatives would argue that they, too, are interested in ideas, not just technical skills. Many of them frame their work around the idea of “computational thinking”—aiming to introduce children to problem-solving strategies that come from the field of computer science but are applicable across many other domains. Learning problem-solving strategies is certainly valuable. But Seymour had a bigger, broader vision. He wanted to support children not only in developing their thinking but also in developing their voice.
西摩认为计算机不仅仅是一种解决问题的工具，更是一种表达媒介。他认为学习编程类似于学习写作，为孩子们提供了组织和表达想法的新方法。西摩希望帮助所有来自不同背景的孩子都有机会表达和分享他们的想法，以便他们能够充分、积极地参与社会生活。
Seymour saw the computer not just as a problem-solving tool but as an expressive medium. He believed that learning to program was analogous to learning to write, providing children with new ways of organizing and expressing their ideas. Seymour wanted to help all children, from all backgrounds, have opportunities to express and share their ideas so that they could be full and active participants in society.
那么，Seymour 会如何看待如今人们对于将人工智能 ( ai )引入K -12 教育的普遍兴奋呢？在《头脑风暴》第 7 章中，Seymour 描述了人工智能研究如何成为他教育和学习工作的重要灵感来源。Seymour 在 1980 年的著作似乎再次具有先见之明。但是，我仍然相信 Seymour 会对当今人工智能教育工作的方式感到非常失望。Seymour 感兴趣的是运用人工智能的理念让孩子们思考自己的思维方式并了解自己的学习方式。当今大多数人工智能教育计划都有非常不同的目标，它们侧重于使用机器智能，而不是理解人类智能。
And what would Seymour think about today’s widespread excitement about introducing artificial intelligence (ai) in k–12 education? In chapter 7 of Mindstorms, Seymour described how ai research was an important source of inspiration for his work in education and learning. Again, Seymour’s writings in 1980 seem prescient. But, still, I believe that Seymour would be very frustrated with the approach of today’s ai-in-education efforts. Seymour was interested in applying ideas from ai to engage children in thinking about their own thinking—and learning about their own learning. Most of today’s ai-in-education initiatives have a very different set of goals, focusing on the use of machine intelligence, rather than the understanding of human intelligence.
那么，是什么限制了西摩思想的传播？为什么他的思想没有产生更大的影响？一个挑战是教育系统普遍抵制变革。但西摩思想的传播、支持和解读方式也存在挑战。例如，西摩经常使用数学和计算机科学中的例子，导致一些人对他的思想的解读过于狭隘，尽管西摩的本意是将他的想法应用于所有学科。西摩经常强调孩子们可以自己创造和学习的例子，导致一些人试图实施他的想法，而没有足够重视教师、家长和同龄人在学习过程中的作用。
So what has limited the spread of Seymour’s ideas? Why haven’t his ideas had an even bigger influence? One challenge is the general resistance to change in educational systems. But there are also challenges in the ways that Seymour’s ideas have been communicated, supported, and interpreted. For instance, Seymour often used examples from mathematics and computer science, leading some people to interpret his ideas too narrowly, even though Seymour intended his ideas to apply across all disciplines. And Seymour often highlighted examples of what children can create and learn on their own, leading some people to try to implement his ideas without paying enough attention to the role of teachers, parents, and peers in the learning process.
当我思考西摩思想的传播时，我喜欢把西摩的思想传播想象成农民播种。有些是数学思想，有些是教育思想，有些是技术思想，有些是认识论思想。西摩的一些思想像野花一样传播到世界各地。有些在一些地方扎根，但在其他地方却没有。他的一些种子仍然潜伏在地下。
When I think about the spread of Seymour’s ideas, I like to think of Seymour planting ideas much as a farmer plants seeds. Some were mathematical ideas, some were pedagogical ideas, some were technological ideas, some were epistemological ideas. Some of Seymour’s ideas spread like wildflowers around the world. Some took root in a few places, but not in others. Some of his seeds still lie dormant in the ground.
我是一个研究人员和教育工作者社区的成员，我们仍然深信西摩的思想和愿景。我们致力于培育西摩播下的种子，并努力为它们提供适当的生长条件。我与西摩密切合作多年，最初是麻省理工学院的研究生，后来成为教职员工。我今天的研究仍然深受西摩思想的影响，我继续探索在不同的学习环境中支持他的想法的方法。作为我工作的框架，我制定了一套四项指导原则，这些原则均受到西摩思想的启发：
I’m part of a community of researchers and educators who continue to believe deeply in Seymour’s ideas and vision. We’re dedicated to nurturing the seeds that Seymour sowed, and we’re trying to provide the right conditions for them to grow. I worked closely with Seymour for many years, first as an mit graduate student and then as a faculty colleague. My research today continues to be deeply influenced by Seymour’s ideas, and I continue to explore ways to support his ideas in different learning contexts. As a framework for my work, I developed a set of four guiding principles that are all inspired by Seymour’s ideas:
项目。西摩极力主张“项目重于问题”。当然，西摩明白解决问题的重要性。但他认为，人们在积极参与有意义的项目时，最有效地学会解决问题（并学习新概念和策略）。学校往往先向学生传授概念，然后才给学生机会参与项目。西摩认为，最好让孩子们通过参与项目来学习新想法，而不是在参与项目之前。
Projects. Seymour provocatively argued for “projects over problems.” Of course, Seymour understood the importance of problem solving. But he believed that people learn to solve problems (and learn new concepts and strategies) most effectively while they are actively engaged in meaningful projects. Too often, schools start by teaching concepts to students, and only then give students a chance to work on projects. Seymour argued that it is best for children to learn new ideas through working on projects, not before working on projects.
激情。在《头脑风暴》的前言中，西摩描述了他童年时对齿轮的迷恋如何为他提供了一种探索重要数学概念的方法。对我来说，前言中最重要、最难忘的一句话是西摩的陈述：“我爱上了齿轮。”西摩明白学习者建立在兴趣和激情之上的重要性。他知道，当人们从事自己热衷的项目时，他们会工作更长时间、更努力，并与想法建立更深入的联系。西摩曾经说过：“教育与解释关系不大，它与参与、与爱上材料有关。”
Passion. In the Preface to Mindstorms, Seymour described how his childhood fascination with gears provided him a way to explore important mathematical concepts. For me, the most important and memorable line in the Preface is Seymour’s statement: “I fell in love with the gears.” Seymour understood the importance of learners building on their interests and passions. He knew that people will work longer and harder, and make deeper connections to ideas, when they’re working on projects that they’re passionate about. Seymour once said: “Education has very little to do with explanation, it has to do with engagement, with falling in love with the material.”
同伴。在《头脑风暴》的最后一章“学习型社会的意象”中，Seymour 写到了巴西的桑巴舞学校，在那里，人们聚集在一起，为一年一度的狂欢节创作音乐和舞蹈套路。最让 Seymour 感兴趣的是桑巴舞学校将不同年龄、不同经验水平的人们聚集在一起：儿童和成人、新手和专家，他们都在一起工作、互相学习。对于 Seymour 来说，这种基于同伴的学习是学习型社会的核心。Seymour 和之前的 Piaget 一样，有时也被批评过于关注个体学习者。但是，他关于桑巴舞学校的著作展示了 Seymour 的另一面。虽然头脑风暴时代的技术还没有为我们今天所看到的基于同伴的在线协作做好准备，但 Seymour 认识到了学习的社会维度的重要性。
Peers. In the final chapter of Mindstorms, titled “Images of the Learning Society,” Seymour wrote about the Brazilian samba schools, where people come together to create music and dance routines for the annual carnival festival. What intrigued Seymour most was the way that samba schools bring together people of all different ages and all different levels of experience: children and adults, novices and experts, all working together, learning with and from one another. For Seymour, this type of peer-based learning was at the core of a Learning Society. Seymour, like Piaget before him, is sometimes criticized for focusing too much on the individual learner. But his writings on samba schools show another side of Seymour. The technologies of the Mindstorms era weren’t quite ready for the type of peer-based online collaboration that we see today, but Seymour recognized the importance of the social dimension of learning.
玩耍。人们通常把玩耍与欢笑和乐趣联系在一起。但对西摩来说，玩耍意味着更多。它涉及试验、冒险、测试界限以及在出现问题时反复调整。西摩有时将这个过程称为“艰苦的乐趣”。他认识到孩子们不希望事情变得容易：他们愿意为他们认为有意义的事情付出巨大的努力。西摩不仅鼓励他人玩耍和艰苦的乐趣；他自己也这样做。他总是在玩弄各种想法、思考各种想法、试验各种想法。我从未见过有人既如此爱玩又对想法如此认真。
Play. Often, people associate play with laughter and fun. But for Seymour, play meant more than that. It involved experimenting, taking risks, testing the boundaries, and iteratively adapting when things go wrong. Seymour sometimes referred to this process as “hard fun.” He recognized that children don’t want things to be easy: They’re willing to work very hard on things that they find meaningful. And Seymour didn’t just encourage play and hard fun for others; he lived it himself. He was always playing with ideas, wrestling with ideas, experimenting with ideas. I never met anyone who was, at once, so playful and so serious about ideas.
西摩不仅仅是一名数学家、教育家、哲学家和计算机科学家，他还是一名活动家。从 20 世纪 40 年代青少年时期在南非与种族隔离作斗争开始，西摩就一直在寻求变革。当他发起新的教育项目时，他经常大喊：“现在是时候了！”他是一个有远大理想、希望实现巨大变革的人。在《头脑风暴》和其他著作中，他提倡革命性地改变我们对儿童、学习和教育的看法。自《头脑风暴》问世以来的四十年里，这些变化更多的是渐进性的，而不是革命性的。西摩的许多梦想都没有实现。
Seymour was not just a mathematician, educator, philosopher, and computer scientist. He was also an activist. From his teenage years in the 1940s, when he battled apartheid in his native South Africa, Seymour was always looking to bring about change. When he initiated new educational projects, he often exclaimed: “Now is the time!” He was a person of big ideas who wanted big changes. In Mindstorms and other writings, he advocated for revolutionary changes in the ways we think about children, learning, and education. In the forty years since Mindstorms, the changes have been more evolutionary than revolutionary. Many of Seymour’s dreams have been unfulfilled.
但我相信，环境正变得越来越肥沃，西摩播下的种子也越来越容易生长。人们越来越认识到，教育体系已经不能满足当今快速变化的社会的需求。越来越多的教育改革者正在倡导与西摩理念相符的变革，比如为孩子们提供更多探索、实验和表达自己的机会，让他们成长为具有创造性思维的人。这些变革是渐进式的，而不是革命性的，但长期趋势正朝着西摩的愿景的方向发展。
But I believe that the environment is becoming more fertile for many of the seeds that Seymour sowed. There is a growing recognition that educational systems are not meeting the needs of today’s fast-changing society. More educational reformers are advocating for changes that are aligned with Seymour’s ideas, such as providing children with more opportunities to explore, experiment, and express themselves, so that they can develop as creative thinkers. The changes are evolutionary, not revolutionary, but the long-term trends are heading in the direction of Seymour’s vision.
阅读《头脑风暴》时，不要被书中描述的 20 世纪 80 年代技术的细节所吸引。相反，要思考如何将西摩的思想融入当今关于教育战略和政策的讨论中。并思考你可以做些什么来培育西摩播下的种子。
As you read Mindstorms, don’t get distracted by the details of the 1980-era technologies that are described in the book. Rather, think about the ways that Seymour’s ideas can be integrated into today’s discussions about educational strategies and policies. And think about what you might do to nurture the seeds that Seymour sowed.
米切尔·雷斯尼克
MITCHEL RESNICK
LEGO Papert 学习研究教授
LEGO Papert Professor of Learning Research
麻省理工学院媒体实验室
MIT Media Lab
前言
PREFACE
我的童年齿轮
The Gears of My Childhood
在我两岁之前，我就已经对汽车产生了浓厚的兴趣。汽车零部件的名称占据了我词汇量的很大一部分：我特别自豪的是，我了解传动系统、变速箱，尤其是差速器的零部件。当然，直到很多年后我才明白齿轮是如何工作的；但一旦我明白了，玩齿轮就成了我最喜爱的消遣。我喜欢以齿轮般的运动将圆形物体相互旋转，自然而然，我的第一个“组装装置”项目是一个粗糙的齿轮系统。
BEFORE I WAS TWO YEARS OLD I HAD DEVELOPED AN INTENSE involvement with automobiles. The names of car parts made up a very substantial portion of my vocabulary: I was particularly proud of knowing about the parts of the transmission system, the gearbox, and most especially the differential. It was, of course, many years later before I understood how gears work; but once I did, playing with gears became a favorite pastime. I loved rotating circular objects against one another in gearlike motions and, naturally, my first “erector set” project was a crude gear system.
我变得擅长在脑海中转动轮子，并建立因果链：“这个轮子这样转，所以那个轮子必须那样转……”我对差速齿轮这样的系统特别感兴趣，它不遵循简单的线性因果链，因为传动轴的运动可以根据两个轮子遇到的阻力以许多不同的方式分配给它们。我清楚地记得，当我发现一个系统可以合法且完全可以理解，而不需要严格确定时，我是多么兴奋。
I became adept at turning wheels in my head and at making chains of cause and effect: “This one turns this way so that must turn that way so…” I found particular pleasure in such systems as the differential gear, which does not follow a simple linear chain of causality since the motion in the transmission shaft can be distributed in many different ways to the two wheels depending on what resistance they encounter. I remember quite vividly my excitement at discovering that a system could be lawful and completely comprehensible without being rigidly deterministic.
我认为微分对我的数学发展比小学所学的任何知识都更有帮助。齿轮作为模型，将许多原本抽象的想法带入我的脑海。我清楚地记得学校数学中的两个例子。我将乘法表视为齿轮，第一次接触两个变量的方程（例如 3 x + 4 y = 10）时，我立即想到了微分。当我在脑海中建立了x和y之间关系的齿轮模型，计算出每个齿轮需要多少齿时，方程已成为我最熟悉的朋友。
I believe that working with differentials did more for my mathematical development than anything I was taught in elementary school. Gears, serving as models, carried many otherwise abstract ideas into my head. I clearly remember two examples from school math. I saw multiplication tables as gears, and my first brush with equations in two variables (e.g., 3x + 4y = 10) immediately evoked the differential. By the time I had made a mental gear model of the relation between x and y, figuring how many teeth each gear needed, the equation had become a comfortable friend.
多年后，当我读到皮亚杰的作品时，这件事让我对他的同化概念有了更深刻的认识，但我立刻意识到，他的讨论并没有完全体现他自己的想法。他几乎完全谈论了同化的认知方面。但其中也包含情感成分。将方程式同化为齿轮当然是一种将旧知识应用于新事物的有效方法。但它的作用不止于此。我确信，这种同化有助于赋予数学一种积极的情感基调，这可以追溯到我幼年时与汽车的接触经历。我相信皮亚杰确实同意这一点。当我与他私下接触时，我明白了他对情感的忽视更多是出于谦虚地认为人们对它知之甚少，而不是出于傲慢地认为它无关紧要。但让我回到我的童年。
Many years later when I read Piaget, this incident served me as a model for his notion of assimilation, except I was immediately struck by the fact that his discussion does not do full justice to his own idea. He talks almost entirely about cognitive aspects of assimilation. But there is also an affective component. Assimilating equations to gears certainly is a powerful way to bring old knowledge to bear on a new object. But it does more as well. I am sure that such assimilations helped to endow mathematics, for me, with a positive affective tone that can be traced back to my infantile experiences with cars. I believe Piaget really agrees. As I came to know him personally I understood that his neglect of the affective comes more from a modest sense that little is known about it than from an arrogant sense of its irrelevance. But let me return to my childhood.
有一天，我惊讶地发现，有些成年人——甚至大多数成年人——不理解甚至不关心齿轮的魔力。我不再过多考虑齿轮，但我从未回避过从这一发现开始的那些问题：为什么对我来说如此简单的事情对其他人来说却难以理解？我骄傲的父亲建议用“聪明”来解释。但我痛苦地意识到，有些人无法理解差速器，却能轻松地做我认为困难得多的事情。慢慢地，我开始形成我仍然认为是学习的基本事实：如果你能将任何事情融入你的模型集合中，那么任何事情都很容易。如果你不能，任何事情都可能非常困难。在这里，我也正在形成一种与皮亚杰产生共鸣的思维方式。对学习的理解必须是遗传的。它必须指知识的起源。一个人能学到什么，以及如何学习，取决于他拥有哪些模型。这又递归地提出了他是如何学习这些模型的问题。因此，“学习规律”必须是关于智力结构如何相互发展，以及在这个过程中如何获得逻辑和情感形式。
One day I was surprised to discover that some adults—even most adults—did not understand or even care about the magic of the gears. I no longer think much about gears, but I have never turned away from the questions that started with that discovery: How could what was so simple for me be incomprehensible to other people? My proud father suggested “being clever” as an explanation. But I was painfully aware that some people who could not understand the differential could easily do things I found much more difficult. Slowly I began to formulate what I still consider the fundamental fact about learning: Anything is easy if you can assimilate it to your collection of models. If you can’t, anything can be painfully difficult. Here too I was developing a way of thinking that would be resonant with Piaget’s. The understanding of learning must be genetic. It must refer to the genesis of knowledge. What an individual can learn, and how he learns it, depends on what models he has available. This raises, recursively, the question of how he learned these models. Thus the “laws of learning” must be about how intellectual structures grow out of one another and about how, in the process, they acquire both logical and emotional form.
本书是应用发生认识论的一次实践，它超越了皮亚杰的认知重点，包含了对情感的关注。它为教育研究开辟了一个新的视角，重点是创造智力模型扎根的条件。在过去的二十年里，我一直在努力做到这一点。在这样做的过程中，我发现自己经常想起我与差速齿轮相遇的几个方面。首先，我记得没有人告诉我要学习差速齿轮。其次，我记得在我与齿轮的关系中，除了理解之外，还有感情、爱。第三，我记得我第一次接触它们是在我第二年。如果任何“科学”教育心理学家试图“衡量”这次遭遇的影响，他可能会失败。它产生了深远的影响，但我猜想，要到很多年后才会显现出来。两岁时的“前后”测试可能会错过它们。
This book is an exercise in an applied genetic epistemology expanded beyond Piaget’s cognitive emphasis to include a concern with the affective. It develops a new perspective for education research focused on creating the conditions under which intellectual models will take root. For the last two decades this is what I have been trying to do. And in doing so I find myself frequently reminded of several aspects of my encounter with the differential gear. First, I remember that no one told me to learn about differential gears. Second, I remember that there was feeling, love, as well as understanding in my relationship with gears. Third, I remember that my first encounter with them was in my second year. If any “scientific” educational psychologist had tried to “measure” the effects of this encounter, he would probably have failed. It had profound consequences but, I conjecture, only very many years later. A “pre- and post-” test at age two would have missed them.
皮亚杰的作品为我提供了一个新的框架来观察我童年时期的齿轮。齿轮可用于说明许多强大的“高级”数学思想，例如群或相对运动。但它的作用不止于此。除了与数学的形式知识相联系外，它还与“身体知识”相联系，即儿童的感觉运动图式。你可以成为齿轮，通过将自己投射到齿轮的位置并随之转动，你可以了解它是如何转动的。正是这种双重关系（既抽象又感觉）赋予了齿轮将强大的数学带入大脑的力量。用我将在后面章节中开发的术语来说，齿轮在这里充当过渡对象。
Piaget’s work gave me a new framework for looking at the gears of my childhood. The gear can be used to illustrate many powerful “advanced” mathematical ideas, such as groups or relative motion. But it does more than this. As well as connecting with the formal knowledge of mathematics, it also connects with the “body knowledge,” the sensorimotor schemata of a child. You can be the gear, you can understand how it turns by projecting yourself into its place and turning with it. It is this double relationship—both abstract and sensory—that gives the gear the power to carry powerful mathematics into the mind. In a terminology I shall develop in later chapters, the gear acts here as a transitional object.
如果一位现代的蒙特梭利老师被我的故事说服了，他可能会建议为孩子们制作一套齿轮。这样，每个孩子都可以拥有我所拥有的体验。但希望如此就会错过故事的本质。我爱上了齿轮。这不能归结为纯粹的“认知”术语。发生了一件非常个人化的事情，我们不能假设它会以完全相同的形式重复发生在其他孩子身上。
A modern-day Montessori might propose, if convinced by my story, to create a gear set for children. Thus every child might have the experience I had. But to hope for this would be to miss the essence of the story. I fell in love with the gears. This is something that cannot be reduced to purely “cognitive” terms. Something very personal happened, and one cannot assume that it would be repeated for other children in exactly the same form.
我的论点可以概括为：齿轮做不到的事情，计算机可以做到。计算机是机器中的普罗透斯。它的本质是它的通用性，它的模拟能力。因为它可以呈现一千种形式，可以实现一千种功能，所以它可以迎合一千种口味。这本书是我过去十年来尝试将计算机变成足够灵活的工具的结果，这样许多孩子都可以自己创造一些像齿轮一样的东西。
My thesis could be summarized as: What the gears cannot do the computer might. The computer is the Proteus of machines. Its essence is its universality, its power to simulate. Because it can take on a thousand forms and can serve a thousand functions, it can appeal to a thousand tastes. This book is the result of my own attempts over the past decade to turn computers into instruments flexible enough so that many children can each create for themselves something like what the gears were for me.
介绍
INTRODUCTION
儿童电脑
Computers for Children
几年前，人们还认为计算机是一种昂贵而奇特的设备。计算机的商业和工业用途影响着普通人，但几乎没有人会想到计算机会成为日常生活的一部分。随着公众开始接受个人计算机的现实，这种观点发生了巨大而迅速的变化。个人计算机体积小巧，价格低廉，可以摆放在每个客厅，甚至每个胸前的口袋里。这类计算机中第一批相当原始的机器的出现足以引起记者的想象，并引发了大量关于未来计算机丰富世界生活的推测性文章。这些文章的主要主题是人们将能够用计算机做什么。大多数作者强调使用计算机玩游戏、娱乐、缴纳所得税、发送电子邮件、购物和银行业务。少数人谈到计算机是一种教学机器。
JUST A FEW YEARS AGO PEOPLE THOUGHT OF COMPUTERS AS EXPENSIVE and exotic devices. Their commercial and industrial uses affected ordinary people, but hardly anyone expected computers to become part of day-to-day life. This view has changed dramatically and rapidly as the public has come to accept the reality of the personal computer, small and inexpensive enough to take its place in every living room or even in every breast pocket. The appearance of the first rather primitive machines in this class was enough to catch the imagination of journalists and produce a rash of speculative articles about life in the computer-rich world to come. The main subject of these articles was what people will be able to do with their computers. Most writers emphasized using computers for games, entertainment, income tax, electronic mail, shopping, and banking. A few talked about the computer as a teaching machine.
本书也提出了个人电脑将做什么的问题，但方式却截然不同。我将讨论电脑如何影响人们的思考和学习方式。我首先通过区分电脑可能增强思考和改变知识获取模式的两种方式来描述我的观点。
This book too poses the question of what will be done with personal computers, but in a very different way. I shall be talking about how computers may affect the way people think and learn. I begin to characterize my perspective by noting a distinction between two ways computers might enhance thinking and change patterns of access to knowledge.
计算机帮助人们思考的工具性用途在科幻小说中得到了戏剧化描述。例如，正如数百万《星际迷航》粉丝所知，星际飞船企业号上有一台计算机，可以快速准确地回答向它提出的复杂问题。但《星际迷航》并没有试图暗示飞船上的人类角色的思维方式与二十世纪人们的思维方式有很大不同。就我们在这些情节中看到的而言，与计算机的接触并没有改变这些人对自己的看法或解决问题的方式。在这本书中，我讨论了计算机的存在不仅在工具上，而且在更本质、概念上对心理过程的贡献，即使人们远离计算机的物理接触，它也会影响他们的思维方式（就像齿轮塑造了我对代数的理解，尽管它们并没有出现在数学课上）。它讲述了一种使科学和技术对绝大多数人来说很陌生的文化的终结。许多文化障碍阻碍了孩子们将科学知识变成自己的知识。在这些障碍中，最明显的是贫困和孤立对身体造成的残酷影响。其他障碍则更具政治性。许多在城市中长大的孩子被科学产品包围，但他们有充分的理由将它们视为“他人”；在许多情况下，它们被视为社会敌人。还有一些障碍更为抽象，但最终具有相同的性质。欧洲和美国最复杂的现代文化的大多数分支都具有强烈的“数学恐惧症”，以至于许多有特权的孩子被有效地（尽管更温和地）阻止将科学当作自己的。在我看来，太空时代的物品，以小型计算机的形式，将跨越这些文化障碍，进入世界各地儿童的私人世界。它们将不仅仅是物理对象。这本书讲述了计算机如何成为强大思想和文化变革种子的载体，它们如何帮助人们与知识建立新的关系，这种关系跨越了人文与科学以及自我知识与人文与科学之间的传统界限。它是关于使用计算机来挑战当前关于谁能在什么年龄理解什么的观念。它是关于使用计算机来质疑发展心理学和能力与态度心理学中的标准假设。它是关于个人计算机及其所处的文化是否将继续成为“工程师”的专属，或者我们是否可以构建智力环境，让今天自认为是“人文主义者”的人们感受到自己是构建计算文化过程的一部分，而不是被孤立。
Instrumental uses of the computer to help people think have been dramatized in science fiction. For example, as millions of “Star Trek” fans know, the starship Enterprise has a computer that gives rapid and accurate answers to complex questions posed to it. But no attempt is made in “Star Trek” to suggest that the human characters aboard think in ways very different from the manner in which people in the twentieth century think. Contact with the computer has not, as far as we are allowed to see in these episodes, changed how these people think about themselves or how they approach problems. In this book I discuss ways in which the computer presence could contribute to mental processes not only instrumentally but in more essential, conceptual ways, influencing how people think even when they are far removed from physical contact with a computer (just as the gears shaped my understanding of algebra although they were not physically present in the math class). It is about an end to the culture that makes science and technology alien to the vast majority of people. Many cultural barriers impede children from making scientific knowledge their own. Among these barriers the most visible are the physically brutal effects of deprivation and isolation. Other barriers are more political. Many children who grow up in our cities are surrounded by the artifacts of science but have good reason to see them as belonging to “the others”; in many cases they are perceived as belonging to the social enemy. Still other obstacles are more abstract, though ultimately of the same nature. Most branches of the most sophisticated modern culture of Europe and the United States are so deeply “mathophobic” that many privileged children are as effectively (if more gently) kept from appropriating science as their own. In my vision, space-age objects, in the form of small computers, will cross these cultural barriers to enter the private worlds of children everywhere. They will do so not as mere physical objects. This book is about how computers can be carriers of powerful ideas and of the seeds of cultural change, how they can help people form new relationships with knowledge that cuts across the traditional lines separating humanities from sciences and knowledge of the self from both of these. It is about using computers to challenge current beliefs about who can understand what and at what age. It is about using computers to question standard assumptions in developmental psychology and in the psychology of aptitudes and attitudes. It is about whether personal computers and the cultures in which they are used will continue to be the creatures of “engineers” alone or whether we can construct intellectual environments in which people who today think of themselves as “humanists” will feel part of, not alienated from, the process of constructing computational cultures.
但是，计算机能做什么与社会选择用它们做什么之间存在着天壤之别。社会有许多方法可以抵制根本性的、具有威胁性的变革。因此，这本书是关于如何面对最终是政治性的选择。它研究了一些变革力量，以及随着计算机开始进入充满政治色彩的教育界，人们对这些力量作出的反应。
But there is a world of difference between what computers can do and what society will choose to do with them. Society has many ways to resist fundamental and threatening change. Thus, this book is about facing choices that are ultimately political. It looks at some of the forces of change and of reaction to those forces that are called into play as the computer presence begins to enter the politically charged world of education.
本书的大部分篇幅都致力于塑造与当前刻板印象截然不同的计算机角色形象。我们所有人，无论是专业人士还是门外汉，都必须有意识地打破我们思考计算机时的习惯。计算技术尚处于起步阶段。如果不将我们今天认为已知的计算机的特性和局限性投射到它们身上，就很难想象未来的计算机。在想象计算机如何进入教育领域时，这一点再正确不过了。我在这里要描绘的儿童与计算机的关系形象远远超出了当今学校的普遍情况，这种说法并不正确。我的形象并没有超越：它朝着相反的方向发展。
Much of the book is devoted to building up images of the role of the computer that are very different from current stereotypes. All of us, professionals as well as laymen, must consciously break the habits we bring to thinking about the computer. Computation is in its infancy. It is hard to think about computers of the future without projecting onto them the properties and the limitations of those we think we know today. And nowhere is this more true than in imagining how computers can enter the world of education. It is not true to say that the image of a child’s relationship with a computer I shall develop here goes far beyond what is common in today’s schools. My image does not go beyond: It goes in the opposite direction.
如今，在很多学校里，“计算机辅助教学”一词意味着让计算机教孩子。有人可能会说，计算机被用来给孩子编程。在我看来，孩子对计算机进行编程，这样，他们既获得了对最现代、最强大的技术的掌握感，又与科学、数学和智力模型构建艺术中的一些最深奥的思想建立了密切的联系。
In many schools today, the phrase “computer-aided instruction” means making the computer teach the child. One might say the computer is being used to program the child. In my vision, the child programs the computer and, in doing so, both acquires a sense of mastery over a piece of the most modern and powerful technology and establishes an intimate contact with some of the deepest ideas from science, from mathematics, and from the art of intellectual model building.
我将描述数百名儿童成为相当高级程序员的学习路径。一旦从正确的角度看待编程，这种情况的发生就不足为奇了。对计算机进行编程无非就是用计算机和人类用户都能“理解”的语言与计算机进行交流。学习语言是孩子们最擅长的事情之一。每个正常的孩子都会学习说话。那么为什么孩子不应该学习与计算机“交谈”呢？
I shall describe learning paths that have led hundreds of children to becoming quite sophisticated programmers. Once programming is seen in the proper perspective, there is nothing very surprising about the fact that this should happen. Programming a computer means nothing more or less than communicating to it in a language that it and the human user can both “understand.” And learning languages is one of the things children do best. Every normal child learns to talk. Why then should a child not learn to “talk” to a computer?
人们认为计算机语言很难的原因有很多。例如，尽管婴儿可以非常轻松地学会说母语，但大多数孩子在学校学习外语时却非常困难，而且他们往往无法成功地学习自己母语的书面版本。学习计算机语言难道不像学习说自己母语那么容易，而更像是学习外语书面语的困难过程吗？大多数人在学习数学时遇到的所有困难难道不会使问题更加复杂吗？
There are many reasons why someone might expect it to be difficult. For example, although babies learn to speak their native language with spectacular ease, most children have great difficulty learning foreign languages in schools and, indeed, often learn the written version of their own language none too successfully. Isn’t learning a computer language more like the difficult process of learning a foreign written language than the easy one of learning to speak one’s own language? And isn’t the problem further compounded by all the difficulties most people encounter learning mathematics?
本书贯穿了两个基本思想。首先，可以设计计算机，使学习与计算机交流成为一个自然的过程，更像是通过生活在法国学习法语，而不是通过美国课堂上不自然的外语教学过程来学习法语。其次，学习与计算机交流可能会改变其他学习方式。计算机可以是一个讲数学和讲字母的实体。我们正在学习如何制造孩子们喜欢与之交流的计算机。当这种交流发生时，孩子们将数学作为一种活的语言来学习。此外，数学交流和字母交流都从对大多数孩子来说陌生而困难的事情转变为自然而简单的事情。与计算机“谈论数学”的想法可以推广到在“数学世界”学习数学的观点——也就是说，在这样的背景下学习数学就像生活在法国学习法语一样。
Two fundamental ideas run through this book. The first is that it is possible to design computers so that learning to communicate with them can be a natural process, more like learning French by living in France than like trying to learn it through the unnatural process of American foreign-language instruction in classrooms. Second, learning to communicate with a computer may change the way other learning takes place. The computer can be a mathematics-speaking and an alphabetic-speaking entity. We are learning how to make computers with which children love to communicate. When this communication occurs, children learn mathematics as a living language. Moreover, mathematical communication and alphabetic communication are thereby both transformed from the alien and therefore difficult things they are for most children into natural and therefore easy ones. The idea of “talking mathematics” to a computer can be generalized to a view of learning mathematics in “Mathland”—that is to say, in a context which is to learning mathematics what living in France is to learning French.
在本书中，数学国度的隐喻将被用来质疑关于人类能力的根深蒂固的假设。人们普遍认为，孩子们要到上学后才能学习正式的几何学，而且大多数人甚至在那时也学不太好。但是，通过对孩子们学习法语的能力提出类似的问题，我们很快就能发现，这些假设是建立在极其薄弱的证据之上的。如果我们必须根据对美国学校孩子们学习法语的糟糕程度的观察来得出我们的结论，我们不得不得出这样的结论：大多数人无法掌握法语。但我们知道，如果所有正常的孩子生活在法国，他们都能很容易地学会法语。我的推测是，当孩子们在不久的将来在计算机丰富的世界中长大时，我们现在认为的过于“正式”或“过于数学化”的很多东西也会同样容易地学会。
In this book the Mathland metaphor will be used to question deeply engrained assumptions about human abilities. It is generally assumed that children cannot learn formal geometry until well into their school years and that most cannot learn it too well even then. But we can quickly see that these assumptions are based on extremely weak evidence by asking analogous questions about the ability of children to learn French. If we had to base our opinions on observation of how poorly children learned French in American schools, we would have to conclude that most people were incapable of mastering it. But we know that all normal children would learn it very easily if they lived in France. My conjecture is that much of what we now see as too “formal” or “too mathematical” will be learned just as easily when children grow up in the computer-rich world of the very near future.
我以考察我们与数学的关系作为主题例子，说明技术和社会过程如何在人类能力观念的构建中相互作用。数学例子也有助于描述学习如何进行以及学习如何出错的理论。
I use the examination of our relationship with mathematics as a thematic example of how technological and social processes interact in the construction of ideas about human capacities. And mathematical examples will also help to describe a theory of how learning works and of how it goes wrong.
我从让·皮亚杰1 那里借用了一个模型，即儿童是他们自己智力结构的建造者。儿童似乎天生就是学习天才，早在他们上学之前就通过一种我称之为“皮亚杰学习”或“无需教导的学习”的过程获得了大量知识。例如，孩子们学习说话，学习在空间中行走所需的直观几何学，学习足够的逻辑和修辞学以应付父母——所有这些都不需要“教导”。我们必须问为什么有些学习发生得如此早和自发，而有些学习却要推迟很多年，或者在没有刻意强加的正式指导的情况下根本不会发生。
I take from Jean Piaget1 a model of children as builders of their own intellectual structures. Children seem to be innately gifted learners, acquiring long before they go to school a vast quantity of knowledge by a process I call “Piagetian learning,” or “learning without being taught.” For example, children learn to speak, learn the intuitive geometry needed to get around in space, and learn enough of logic and rhetorics to get around parents—all this without being “taught.” We must ask why some learning takes place so early and spontaneously while some is delayed many years or does not happen at all without deliberately imposed formal instruction.
如果我们真正将“儿童作为建造者”来看待，我们就会找到答案。所有建造者都需要材料来建造。我与皮亚杰的观点不一致的地方在于，我认为周围的文化是这些材料的来源。在某些情况下，文化提供了丰富的材料，从而促进了建设性的皮亚杰式学习。例如，许多重要的东西（刀和叉、母亲和父亲、鞋子和袜子）都是成对出现的，这是构建直觉数字感的“材料”。但在许多情况下，皮亚杰会用概念的复杂性或形式性来解释某个概念发展较慢的原因，而我认为关键因素是文化中那些使概念简单而具体的材料相对匮乏。在其他情况下，文化可能会提供材料，但阻碍了它们的使用。在形式数学中，既存在形式材料的短缺，也存在文化障碍。当代文化中普遍存在的对数学的恐惧阻碍了许多人学习他们所认为的“数学”的任何知识，尽管他们可能对他们不认为是数学的知识没有任何困难。
If we really look at the “child as builder,” we are on our way to an answer. All builders need materials to build with. Where I am at variance with Piaget is in the role I attribute to the surrounding cultures as a source of these materials. In some cases the culture supplies them in abundance, thus facilitating constructive Piagetian learning. For example, the fact that so many important things (knives and forks, mothers and fathers, shoes and socks) come in pairs is a “material” for the construction of an intuitive sense of number. But in many cases where Piaget would explain the slower development of a particular concept by its greater complexity or formality, I see the critical factor as the relative poverty of the culture in those materials that would make the concept simple and concrete. In yet other cases, the culture may provide materials but block their use. In the case of formal mathematics, there is both a shortage of formal materials and a cultural block as well. The mathophobia endemic in contemporary culture blocks many people from learning anything they recognize as “math,” although they may have no trouble with mathematical knowledge they do not perceive as such.
我们将一次又一次地看到，数学恐惧症的后果远远不止阻碍数学和科学的学习。它们与其他地方性的“文化毒素”相互作用，例如，与流行的能力理论相互作用，污染了人们作为学习者的自我形象。学校数学学习困难往往是侵入性智力过程的第一步，这一过程导致我们所有人都将自己定义为各种能力和无能，比如“数学”或“不是数学”、“艺术”或“不是艺术”、“音乐”或“不是音乐”、“深刻”或“肤浅”、“聪明”或“愚蠢”。因此，缺陷成为身份，学习从幼儿自由探索世界变成了一项充满不安全感和自我限制的苦差事。
We shall see again and again that the consequences of mathophobia go far beyond obstructing the learning of mathematics and science. They interact with other endemic “cultural toxins,” for example, with popular theories of aptitudes, to contaminate peoples’ images of themselves as learners. Difficulty with school math is often the first step of an invasive intellectual process that leads us all to define ourselves as bundles of aptitudes and ineptitudes, as being “mathematical” or “not mathematical,” “artistic” or “not artistic,” “musical” or “not musical,” “profound” or “superficial,” “intelligent” or “dumb.” Thus deficiency becomes identity and learning is transformed from the early child’s free exploration of the world to a chore beset by insecurities and self-imposed restrictions.
两个主要主题——儿童可以熟练地学习使用计算机，以及学习使用计算机可以改变他们学习其他一切的方式——塑造了我对计算机和教育的研究议程。在过去十年中，我有幸与麻省理工学院的一组同事和学生（人工智能实验室的LOGO 2小组）合作，创造了儿童可以学习与计算机交流的环境。模仿儿童学习说话的方式的隐喻一直伴随着我们，并导致了与传统截然不同的教育和教育研究愿景。对于从事教师职业的人来说，“教育”一词往往会让人联想到“教学”，尤其是课堂教学。因此，教育研究的目标往往集中在如何改进课堂教学上。但是，如果如我在这里强调的那样，成功学习的模式是儿童学习说话的方式，这个过程不需要刻意和有组织的教学，那么目标就大不相同了。我认为教室是一种人为的、低效的学习环境，社会被迫发明这种环境，因为非正式环境在某些基本学习领域（如写作、语法或学校数学）上存在缺陷。我相信，计算机的存在将使我们能够改变教室外的学习环境，以至于目前学校试图教授的知识中，大部分（如果不是全部的话）知识都将在孩子学习说话时轻松、成功地学习，而无需有组织的指导。这显然意味着我们今天所知道的学校在未来将不复存在。但它们是否会通过转型为新事物来适应，还是会逐渐消亡并被取代，这仍是一个悬而未决的问题。
Two major themes—that children can learn to use computers in a masterful way, and that learning to use computers can change the way they learn everything else—have shaped my research agenda on computers and education. Over the past ten years I have had the good fortune to work with a group of colleagues and students at MIT (the LOGO2 group in the Artificial Intelligence Laboratory) to create environments in which children can learn to communicate with computers. The metaphor of imitating the way the child learns to talk has been constantly with us in this work and has led to a vision of education and of education research very different from the traditional ones. For people in the teaching professions, the word “education” tends to evoke “teaching,” particularly classroom teaching. The goal of education research tends therefore to be focused on how to improve classroom teaching. But if, as I have stressed here, the model of successful learning is the way a child learns to talk, a process that takes place without deliberate and organized teaching, the goal set is very different. I see the classroom as an artificial and inefficient learning environment that society has been forced to invent because its informal environments fail in certain essential learning domains, such as writing or grammar or school math. I believe that the computer presence will enable us to so modify the learning environment outside the classrooms that much if not all the knowledge schools presently try to teach with such pain and expense and such limited success will be learned, as the child learns to talk, painlessly, successfully, and without organized instruction. This obviously implies that schools as we know them today will have no place in the future. But it is an open question whether they will adapt by transforming themselves into something new or wither away and be replaced.
尽管技术将在实现我对未来教育的愿景中发挥重要作用，但我关注的重点不是机器，而是思维，尤其是智力运动和文化如何定义自己和发展。事实上，我赋予计算机的角色是文化“病菌”或“种子”的载体，其智力产品一旦在积极成长的头脑中扎根，就不需要技术支持。许多（如果不是全部）在成长过程中热爱数学并有数学天赋的孩子，至少部分地将这种感觉归功于这样一个事实：他们碰巧从成年人那里获得了“数学文化”的“病菌”，可以说，成年人知道如何讲数学，即使只是像莫里哀让乔丹先生在不知不觉中讲散文一样。这些“讲数学”的成年人不一定知道如何解方程；相反，他们思维的转变体现在他们论证的逻辑性上，以及对他们来说，玩耍往往就是玩谜题、双关语和悖论之类的东西。那些不愿意接受数学和科学教育的孩子中，有很多人的成长环境恰好缺少会讲数学的成年人。这些孩子来到学校时，缺乏轻松学习学校数学所必需的要素。学校无法提供这些缺失的要素，而且，通过强迫孩子们进入注定要失败的学习环境，学校会在孩子心中产生强烈的负面情绪，对数学，甚至对学习总体产生负面情绪。这样就形成了一个恶性循环。因为这些孩子有朝一日会为人父母，他们不仅无法把数学的细菌传给他们的孩子，而且几乎肯定会把相反的、具有智力破坏性的数学恐惧症细菌传染给他们的孩子。
Although technology will play an essential role in the realization of my vision of the future of education, my central focus is not on the machine but on the mind, and particularly on the way in which intellectual movements and cultures define themselves and grow. Indeed, the role I give to the computer is that of a carrier of cultural “germs” or “seeds” whose intellectual products will not need technological support once they take root in an actively growing mind. Many if not all the children who grow up with a love and aptitude for mathematics owe this feeling, at least in part, to the fact that they happened to acquire “germs” of the “math culture” from adults, who, one might say, knew how to speak mathematics, even if only in the way that Moliere had M. Jourdain speak prose without knowing it. These “math-speaking” adults do not necessarily know how to solve equations; rather, they are marked by a turn of mind that shows up in the logic of their arguments and in the fact that for them to play is often to play with such things as puzzles, puns, and paradoxes. Those children who prove recalcitrant to math and science education include many whose environments happened to be relatively poor in math-speaking adults. Such children come to school lacking elements necessary for the easy learning of school math. School has been unable to supply these missing elements, and, by forcing the children into learning situations doomed in advance, it generates powerful negative feelings about mathematics and perhaps about learning in general. Thus is set up a vicious self-perpetuating cycle. For these same children will one day be parents and will not only fail to pass on mathematical germs but will almost certainly infect their children with the opposing and intellectually destructive germs of mathophobia.
幸运的是，只要在某一时刻打破自我延续的循环，它就足以永远被打破。我将展示计算机如何使我们能够做到这一点，从而打破恶性循环而不产生对机器的依赖。我的讨论与大多数关于“先天与后天”的争论有两点不同。我将更具体地说明智力成长需要哪些类型的养育，以及如何在家庭和更广泛的社会背景下创造这种养育。
Fortunately it is sufficient to break the self-perpetuating cycle at one point for it to remain broken forever. I shall show how computers might enable us to do this, thereby breaking the vicious cycle without creating a dependence on machines. My discussion differs from most arguments about “nature versus nurture” in two ways. I shall be much more specific both about what kinds of nurturance are needed for intellectual growth and about what can be done to create such nurturance in the home as well as in the wider social context.
因此，这本书实际上是在讲述一种文化、一种思维方式、一种观念是如何在年轻人的头脑中扎根的。我怀疑过于抽象地思考这些问题，因此我将在此以特别有限的焦点进行写作。事实上，我将集中讨论我最了解的那些思维方式。我首先从我对自己发展的了解开始。我非常谦虚地这样做，并不暗示我已经成为每个人都应该成为的人。但我认为理解学习的最好方法是首先理解特定的、精心挑选的案例，然后再考虑如何从这种理解中概括出来。如果你不去思考某件事，你就无法认真思考思考。而我最了解如何思考的事情就是数学。当我在这本书中写数学时，我并不认为自己是在为那些对数学思维本身感兴趣的数学家写作。我感兴趣的是人们如何思考以及如何学习思考的普遍问题。
Thus this book is really about how a culture, a way of thinking, an idea comes to inhabit a young mind. I am suspicious of thinking about such problems too abstractly, and I shall write here with particular restricted focus. I shall in fact concentrate on those ways of thinking that I know best. I begin by looking at what I know about my own development. I do this in all humility, without any implication that what I have become is what everyone should become. But I think that the best way to understand learning is first to understand specific, well-chosen cases and then to worry afterward about how to generalize from this understanding. You can’t think seriously about thinking without thinking about thinking about something. And the something I know best how to think about is mathematics. When in this book I write of mathematics, I do not think of myself as writing for an audience of mathematicians interested in mathematical thinking for its own sake. My interest is in universal issues of how people think and how they learn to think.
当我回想自己成为数学家的过程时，我发现很多东西都很独特，很多东西无法作为教育改革的普遍愿景的一部分进行复制。我当然不认为我们希望每个人都成为数学家。但我认为，我从数学中获得的乐趣应该成为教育普遍愿景的一部分。如果我们能够掌握一个人的经历的本质，我们也许能够以其他方式复制其结果，特别是在抽象事物中发现美的结果。所以我会写很多关于数学的文章。我向讨厌数学的读者表示歉意，但我同时提出要帮助他们学会更好地喜欢数学——或者至少改变他们对“谈论数学”的印象。
When I trace how I came to be a mathematician, I see much that was idiosyncratic, much that could not be duplicated as part of a generalized vision of education reform. And I certainly don’t think that we would want everyone to become a mathematician. But I think that the kind of pleasure I take in mathematics should be part of a general vision of what education should be about. If we can grasp the essence of one person’s experiences, we may be able to replicate its consequences in other ways, and in particular this consequence of finding beauty in abstract things. And so I shall be writing quite a bit about mathematics. I give my apologies to readers who hate mathematics, but I couple that apology with an offer to help them learn to like it a little better—or at least to change their image of what “speaking mathematics” can be all about.
在本书的前言中，我描述了齿轮如何帮助数学思想进入我的生活。有几个特点促成了它们的有效性。首先，它们是我自然“风景”的一部分，嵌入我周围的文化中。这使我能够自己找到它们并以自己的方式与它们联系起来。其次，齿轮是我周围成年人世界的一部分，通过它们，我可以与这些人产生联系。第三，我可以用我的身体来思考齿轮。我可以通过想象身体转动来感受齿轮是如何转动的。这使我能够利用我的“身体知识”来思考齿轮系统。最后，因为从非常真实的意义上讲，齿轮之间的关系包含了大量数学信息，所以我可以用齿轮来思考形式系统。我描述了齿轮作为“思考对象”的方式。在我成长为数学家的过程中，我让它们成为我自己的思考对象。在我作为一名教育研究者的工作中，齿轮也成为了我的思考对象。我的目标是设计其他物品，让孩子们可以自己制作，并以自己的方式制作。本书的大部分内容将描述我进行此类研究的过程。我首先描述一个构造的计算“思考对象”的例子。这就是“海龟” 。3
In the Foreword of this book I described how gears helped mathematical ideas to enter my life. Several qualities contributed to their effectiveness. First, they were part of my natural “landscape,” embedded in the culture around me. This made it possible for me to find them myself and relate to them in my own fashion. Second, gears were part of the world of adults around me, and through them I could relate to these people. Third, I could use my body to think about the gears. I could feel how gears turn by imagining by body turning. This made it possible for me to draw on my “body knowledge” to think about gear systems. And finally, because, in a very real sense, the relationship between gears contains a great deal of mathematical information, I could use the gears to think about formal systems. I have described the way in which the gears served as an “object-to-think-with.” I made them that for myself in my own development as a mathematician. The gears have also served me as an object-to-think-with in my work as an educational researcher. My goal has been the design of other objects that children can make theirs for themselves and in their own ways. Much of this book will describe my path through this kind of research. I begin by describing one example of a constructed computational “object-to-think-with.” This is the “Turtle.”3
本书中乌龟的核心作用并不意味着我将它视为解决所有教育问题的灵丹妙药。我认为它是一种有价值的教育对象，但它的主要作用是作为其他尚未发明的对象的一个模型。我感兴趣的是“思考对象”的发明过程，这些对象中存在着文化存在、嵌入知识和个人认同可能性的交集。
The central role of the Turtle in this book should not be taken to mean that I propose it as a panacea for all educational problems. I see it as a valuable educational object, but its principal role here is to serve as a model for other objects, yet to be invented. My interest is in the process of invention of “objects-to-think-with,” objects in which there is an intersection of cultural presence, embedded knowledge, and the possibility for personal identification.
乌龟是一种计算机控制的机械动物。它存在于“ LOGO环境”的认知微型文化中，LOGO是与乌龟进行通信的计算机语言。乌龟除了便于编程和思考之外，没有其他用途。有些乌龟是抽象的对象，存在于计算机屏幕上。其他的，比如扉页上展示的地面乌龟，是可以像任何机械玩具一样拿起的物理对象。第一次接触乌龟时，通常先向孩子展示如何通过在键盘上输入命令来移动乌龟。FORWARD 100使乌龟沿直线移动 100 个乌龟步，每个步长约为 1 毫米。输入RIGHT 90 使乌龟在原地旋转 90 度。输入PENDOWN使乌龟放下笔，以留下可见的路径痕迹，而PENUP指示它举起笔。当然，孩子需要进行大量探索才能掌握数字的含义。但这个任务足够吸引人，可以帮助大多数孩子完成这个学习过程。
The Turtle is a computer-controlled cybernetic animal. It exists within the cognitive minicultures of the “LOGO environment,” LOGO being the computer language in which communication with the Turtle takes place. The Turtle serves no other purpose than of being good to program and good to think with. Some Turtles are abstract objects that live on computer screens. Others, like the floor Turtles shown in the frontispiece are physical objects that can be picked up like any mechanical toy. A first encounter often begins by showing the child how a Turtle can be made to move by typing commands at a keyboard. FORWARD 100 makes the Turtle move in a straight line a distance of 100 Turtle steps of about a millimeter each. Typing RIGHT 90 causes the Turtle to pivot in place through 90 degrees. Typing PENDOWN causes the Turtle to lower a pen so as to leave a visible trace of its path while PENUP instructs it to raise the pen. Of course the child needs to explore a great deal before gaining mastery of what the numbers mean. But the task is engaging enough to carry most children through this learning process.
编程的概念是通过教 Turtle 一个新单词的比喻来介绍的。这很简单，孩子们通常通过绘制适当的形状来开始他们的编程体验，即对 Turtle 进行编程，使其响应孩子发明的新命令，例如SQUARE或TRIANGLE或SQ或TRI或孩子想要的任何命令。一旦定义了新命令，就可以用来定义其他命令。例如，正如图 1中的房子是由三角形和正方形组成的，绘制它的程序也是由绘制正方形和三角形的命令组成的。图 1显示了该程序演变的四个步骤。从这些简单的绘图开始，年轻的程序员可以朝许多不同的方向前进。有些人致力于绘制更复杂的图形或抽象图形。有些人不再将 Turtle 用作绘图工具，而是学习使用其触摸传感器对其进行编程以寻找或避开物体。4后来，孩子们了解到，可以对计算机进行编程以制作音乐以及移动 Turtle，并通过对 Turtle 进行编程使其跳舞来将这两项活动结合起来。或者，他们可以从地面上的海龟游戏过渡到“屏幕海龟游戏”，通过编程让海龟以鲜艳的色彩绘制动态图像。这些例子千差万别，但孩子们在每一个例子中都在学习如何控制一个异常丰富和复杂的“微观世界”。
The idea of programming is introduced through the metaphor of teaching the Turtle a new word. This is simply done, and children often begin their programming experience by programming the Turtle to respond to new commands invented by the child such as SQUARE or TRIANGLE or SQ or TRI or whatever the child wishes, by drawing the appropriate shapes. New commands once defined can be used to define others. For example just as the house in Figure 1 is built out of a triangle and a square, the program for drawing it is built out of the commands for drawing a square and a triangle. Figure 1 shows four steps in the evolution of this program. From these simple drawings the young programmer can go on in many different directions. Some work on more complex drawings, either figural or abstract. Some abandon the use of the Turtle as a drawing instrument and learn to use its touch sensors to program it to seek out or avoid objects.4 Later children learn that the computer can be programmed to make music as well as move Turtles and combine the two activities by programming Turtles to dance. Or they can move on from floor Turtles to “screen Turtles,” which they program to draw moving pictures in bright colors. The examples are infinitely varied, but in each the child is learning how to exercise control over an exceptionally rich and sophisticated “micro-world.”
从未见过交互式计算机显示器的读者可能很难想象这会带来什么。作为一种脑力锻炼，他们可能会想象一个电子画板，一个不太遥远的未来的计算机图形显示器。这是一个可以显示彩色动态图像的电视屏幕。您还可以在其上“绘图”，给它下达指令，可能是通过打字、说话，也可能是用魔杖指向。根据要求，屏幕上会出现一个调色板。您可以通过用魔杖指向它来选择一种颜色。直到您改变选择，魔杖都会以该颜色绘制。到目前为止，与传统艺术材料的区别似乎很小，但当您开始考虑编辑绘图时，这种区别就变得非常真实。您可以用计算机语言“与您的绘图对话”。您可以“告诉”它用那种颜色替换这种颜色。或者让绘图动起来。或者复制两份并让它们反向旋转。或者用声音调色板替换调色板并“绘制”一段音乐。您可以将您的工作存档在计算机内存中并随时检索，或者将其传送到连接到中央通信网络的数百万台其他计算机中的任一台的内存中，以供您的朋友享用。
Readers who have never seen an interactive computer display might find it hard to imagine where this can lead. As a mental exercise they might like to imagine an electronic sketchpad, a computer graphics display of the not-too-distant future. This is a television screen that can display moving pictures in color. You can also “draw” on it, giving it instructions, perhaps by typing, perhaps by speaking, or perhaps by pointing with a wand. On request, a palette of colors could appear on the screen. You can choose a color by pointing at it with the wand. Until you change your choice, the wand draws in that color. Up to this point the distinction from traditional art materials may seem slight, but the distinction becomes very real when you begin to think about editing the drawing. You can “talk to your drawing” in computer language. You can “tell” it to replace this color with that. Or set a drawing in motion. Or make two copies and set them in counterrotating motion. Or replace the color palette with a sound palette and “draw” a piece of music. You can file your work in computer memory and retrieve it at your pleasure or have it delivered into the memory of any of the many millions of other computers linked to the central communication network for the pleasure of your friends.
这一切都很有趣，这一点毋庸置疑。但这不仅仅是有趣。孩子们正在经历非常有效的学习。使用电子画板的孩子们正在学习一种语言，用于谈论形状和形状的流动、速度和变化率、过程和程序。他们正在学习讲数学，并树立了自己作为数学家的新形象。
That all this would be fun needs no argument. But it is more than fun. Very powerful kinds of learning are taking place. Children working with an electronic sketchpad are learning a language for talking about shapes and fluxes of shapes, about velocities and rates of change, about processes and procedures. They are learning to speak mathematics and acquiring a new image of themselves as mathematicians.
在描述孩子们与海龟一起工作时，我暗示孩子们可以学习编程。对于一些读者来说，这可能相当于我们走进剧院看戏剧时需要暂停怀疑。对他们来说，编程是一项复杂且有市场价值的技能，需要一些数学天赋的成年人才能掌握。但我的经历却大不相同。我见过数百名小学生非常轻松地学会编程，而且越来越多的证据表明，年龄小得多的孩子也可以做到。这些研究中的孩子并不出类拔萃，或者更确切地说，他们在各个方面都很出类拔萃。有些孩子在学校里非常成功，有些被诊断为情感或认知障碍。有些孩子患有严重的脑瘫，他们从未有意识地操纵过物体。他们中的一些人以“数学”形式表达自己的才能，有些人以“语言”形式表达，有些人以艺术“视觉”或“音乐”形式表达。
In my description of children working with Turtles, I implied that children can learn to program. For some readers this might be tantamount to the suspension of disbelief required when we enter a theater to watch a play. For them programming is a complex and marketable skill acquired by some mathematically gifted adults. But my experience is very different. I have seen hundreds of elementary schoolchildren learn very easily to program, and evidence is accumulating to indicate that much younger children could do so as well. The children in these studies are not exceptional, or rather, they are exceptional in every conceivable way. Some of the children were highly successful in school, some were diagnosed as emotionally or cognitively disabled. Some of the children were so severely afflicted by cerebral palsy that they had never purposefully manipulated physical objects. Some of them had expressed their talents in “mathematical” forms, some in “verbal” forms, and some in artistically “visual” or in “musical” forms.
图 1
Figure 1
当然，这些孩子的编程能力并没有达到与口语能力相当的水平。如果我们认真对待 Mathland 的比喻，那么他们的计算机体验更像是在法国度假一两周来学习法语，而不是在那里生活。但就像和说外语的表兄弟一起度假的孩子一样，他们显然正在走向“会说计算机”的路上。
Of course these children did not achieve a fluency in programming that came close to matching their use of spoken language. If we take the Mathland metaphor seriously, their computer experience was more like learning French by spending a week or two on vacation in France than like living there. But like children who have spent a vacation with foreign-speaking cousins, they were clearly on their way to “speaking computer.”
当我思考这些研究的意义时，我有两个清晰的印象。首先，在适当的条件下，所有儿童都将熟练掌握编程，这将使其成为他们更高级的智力成就之一。其次，“适当的条件”与现在已成为学校常态的计算机使用方式截然不同。我将在本书中讨论的与计算机建立关系的必要条件要求比教育规划者目前预期的更多、更自由地使用计算机。他们需要一种计算机语言和一种围绕该语言的学习环境，这与学校目前提供的非常不同。他们甚至需要一种与学校目前购买的计算机截然不同的计算机。
When I have thought about what these studies mean, I am left with two clear impressions. First, that all children will, under the right conditions, acquire a proficiency with programming that will make it one of their more advanced intellectual accomplishments. Second, that the “right conditions” are very different from the kind of access to computers that is now becoming established as the norm in schools. The conditions necessary for the kind of relationships with a computer that I will be writing about in this book require more and freer access to the computer than educational planners currently anticipate. And they require a kind of computer language and a learning environment around that language very different from those the schools are now providing. They even require a kind of computer rather different from those that the schools are currently buying.
本书的大部分篇幅都用来传达一些信息，即计算机、计算机语言以及更广泛意义上的计算机文化之间的选择，这些选择会影响孩子们从计算中学习到的程度以及他们从中得到的好处。但是，每个孩子免费使用计算机的经济可行性问题可以立即得到解决。我希望通过这样做消除读者对我一直在谈论的“教育愿景”的“经济现实性”的任何疑虑。
It will take most of this book for me to convey some sense of the choices among computers, computer languages, and more generally, among computer cultures, that influence how well children will learn from working with computation and what benefits they will get from doing so. But the question of the economic feasibility of free access to computers for every child can be dealt with immediately. In doing so I hope to remove any doubts readers may have about the “economic realism” of the “vision of education” I have been talking about.
我对新型学习环境的设想要求孩子们可以自由接触电脑。这可能因为孩子的家人买了一台电脑或者孩子的朋友有一台电脑而实现。为了便于讨论（并将我们的讨论扩展到所有社会群体），我们假设这种情况的发生是因为学校为每个学生提供了一台功能强大的个人电脑。大多数“务实”的人（包括家长、教师、校长和基金会管理者）对这个想法的反应大致相同：“即使电脑可以产生你所说的所有效果，你的想法仍然不可能付诸实践。钱从哪里来？”
My vision of a new kind of learning environment demands free contact between children and computers. This could happen because the child’s family buys one or a child’s friends have one. For purposes of discussion here (and to extend our discussion to all social groups) let us assume that it happens because schools give every one of their students his or her own powerful personal computer. Most “practical” people (including parents, teachers, school principals, and foundation administrators) react to this idea in much the same way: “Even if computers could have all the effects you talk about, it would still be impossible to put your ideas into action. Where would the money come from?”
我们必须正视这些人的言论。他们错了。让我们考虑一下 1987 年进入幼儿园的那批孩子，即“2000 届”，并进行一些计算。如今，从幼儿园到十二年级，对一个孩子进行十三年教育的直接公共成本超过 20,000 美元（而对于 2000 届，可能接近 30,000 美元）。保守估计，为每个孩子提供一台性能足以满足本书所述教育目的的个人计算机，并在必要时对其进行升级、维修和更换，每名学生的成本约为 1,000 美元，分摊到十三年的学校教育中。因此，2000 届学生的“计算机成本”仅占公共教育总支出的 5% 左右，即使教育成本结构的其他部分没有因为计算机的出现而发生变化，情况也是如此。但事实上，计算机在教育领域很有可能降低教育的其他方面。学校可能能够将其周期从 13 年缩短到 12 年；他们可能能够利用计算机赋予学生的更大自主权，将班级规模增加一到两名学生，而不会减少对每个学生的个人关注。这两种举措中的任何一种都可以“收回”计算机成本。
What these people are saying needs to be faced squarely. They are wrong. Let’s consider the cohort of children who will enter kindergarten in the year 1987, the “Class of 2000,” and let’s do some arithmetic. The direct public cost of schooling a child for thirteen years, from kindergarten through twelfth grade is over $20,000 today (and for the class of 2000, it may be closer to $30,000). A conservatively high estimate of the cost of supplying each of these children with a personal computer with enough power for it to serve the kinds of educational ends described in this book, and of upgrading, repairing, and replacing it when necessary would be about $1,000 per student, distributed over thirteen years in school. Thus, “computer costs” for the class of 2000 would represent only about 5 percent of the total public expenditure on education, and this would be the case even if nothing else in the structure of educational costs changed because of the computer presence. But in fact computers in education stand a good chance of making other aspects of education cheaper. Schools might be able to reduce their cycle from thirteen years to twelve years; they might be able to take advantage of the greater autonomy the computer gives students and increase the size of classes by one or two students without decreasing the personal attention each student is given. Either of these two moves would “recuperate” the computer cost.
我的目标不是教育节约：不是利用计算来减少孩子在一所没有变化的学校里学习一年的时间，也不是将额外的孩子送入小学课堂。这个教育“预算平衡”小练习的目的是在读者阅读本书第一章时对他们的心态产生一些影响。我将自己描述为教育乌托邦主义者——不是因为我预测了教育的未来，其中孩子们被高科技包围，而是因为我相信，某些非常强大的计算技术和计算思想的用途可以为孩子们提供学习、思考和情感及认知成长的新可能性。在接下来的章节中，我将尝试让您了解这些可能性，其中许多都依赖于计算机丰富的未来，计算机将成为每个孩子生活中的重要组成部分。但我希望我的读者非常清楚，我愿景中和本书中的“乌托邦”是一种使用计算机的特殊方式，是在计算机和人之间建立新的关系的方式——计算机将存在并为人们所用，这只是一个保守的前提。
My goal is not educational economies: It is not to use computation to shave a year off the time a child spends in an otherwise unchanged school or to push an extra child into an elementary school classroom. The point of this little exercise in educational “budget balancing” is to do something to the state of mind of my readers as they turn to the first chapter of this book. I have described myself as an educational utopian—not because I have projected a future of education in which children are surrounded by high technology, but because I believe that certain uses of very powerful computational technology and computational ideas can provide children with new possibilities for learning, thinking, and growing emotionally as well as cognitively. In the chapters that follow, I shall try to give you some idea of these possibilities, many of which are dependent on a computer-rich future, a future where a computer will be a significant part of every child’s life. But I want my readers to be very clear that what is “utopian” in my vision and in this book is a particular way of using computers, of forging new relationships between computers and people—that the computer will be there to be used is simply a conservative premise.
第一章
CHAPTER 1
计算机和计算机文化
Computers and Computer Cultures
在大多数当代教育环境中，孩子们接触计算机时，计算机被用来测试孩子们的能力，提供适当难度的练习，提供反馈和传递信息。计算机为孩子编程。在LOGO环境中，这种关系是相反的。孩子，甚至在学龄前，都是控制者：孩子为计算机编程。在教计算机如何思考时，孩子们开始探索他们自己的思考方式。这种体验可能令人兴奋：思考思考会让孩子变成一个认识论者，这是大多数成年人都没有的经历。
IN MOST CONTEMPORARY EDUCATIONAL SITUATIONS WHERE CHILDREN come into contact with computers, the computer is used to put children through their paces, to provide exercises of an appropriate level of difficulty, to provide feedback, and to dispense information. The computer programming the child. In the LOGO environment the relationship is reversed. The child, even at preschool ages, is in control: The child programs the computer. And in teaching the computer how to think, children embark on an exploration about how they themselves think. The experience can be heady: Thinking about thinking turns the child into an epistemologist, an experience not even shared by most adults.
当我与皮亚杰一起工作时，儿童作为认识论者的这种强有力的形象引起了我的想象。1964 年，在日内瓦皮亚杰发生认识论中心工作了五年后，我对他将儿童视为自己知识结构的积极建设者的方式印象深刻。但是，说知识结构是由学习者构建而不是由教师教授并不意味着它们是从无到有构建的。相反：与其他建设者一样，儿童会根据自己的需要使用他们找到的材料，最突出的是周围文化所暗示的模型和隐喻。
This powerful image of child as epistemologist caught my imagination while I was working with Piaget. In 1964, after five years at Piaget’s Center for Genetic Epistemology in Geneva, I came away impressed by his way of looking at children as the active builders of their own intellectual structures. But to say that intellectual structures are built by the learner rather than taught by a teacher does not mean that they are built from nothing. On the contrary: Like other builders, children appropriate to their own use materials they find about them, most saliently the models and metaphors suggested by the surrounding culture.
皮亚杰曾写到儿童发展不同智力的顺序。我比他更看重特定文化提供的材料对确定该顺序的影响。例如，我们的文化中非常丰富的材料，这些材料对儿童构建某些数字和逻辑思维组成部分非常有用。孩子们学会了数数；他们学会了数数的结果与顺序和特殊排列无关；他们将这种“守恒定律”扩展到思考液体在倒出时的特性以及固体改变形状的特性。孩子们在潜意识中和“自发地”发展这些思维成分，也就是说，没有经过刻意的教导。知识的其他组成部分，例如进行排列和组合的技能，发展得更慢，或者如果没有正规的学校教育，根本不会发展。总的来说，这本书的论点是，在许多重要情况下，这种发展差异可以归因于我们的文化在材料方面相对贫乏，而这些材料可以用来构建看似“更先进”的智力结构。这一论点与皮亚杰的文化解释截然不同，皮亚杰的文化解释寻求的是欧洲或美国城市儿童与非洲丛林部落儿童之间的差异。当我在这里谈到“我们的”文化时，我指的是一些不那么狭隘的东西。我并不是想把纽约与乍得进行对比。我感兴趣的是前计算机文化（无论是在美国城市还是非洲部落）与未来几十年可能在各地发展的“计算机文化”之间的差异。
Piaget writes about the order in which the child develops different intellectual abilities. I give more weight than he does to the influence of the materials a particular culture provides in determining that order. For example, our culture is very rich in materials useful for the child’s construction of certain components of numerical and logical thinking. Children learn to count; they learn that the result of counting is independent of order and special arrangement; they extend this “conservation” to thinking about the properties of liquids as they are poured and of solids which change their shape. Children develop these components of thinking preconsciously and “spontaneously,” that is to say without deliberate teaching. Other components of knowledge, such as the skills involved in doing permutations and combinations, develop more slowly, or do not develop at all without formal schooling. Taken as a whole, this book is an argument that in many important cases this developmental difference can be attributed to our culture’s relative poverty in materials from which the apparently “more advanced” intellectual structures can be built. This argument will be very different from cultural interpretations of Piaget that look for differences between city children in Europe or the United States and tribal children in African jungles. When I speak here of “our” culture I mean something less parochial. I am not trying to contrast New York with Chad. I am interested in the difference between precomputer cultures (whether in American cities or African tribes) and the “computer cultures” that may develop everywhere in the next decades.
我已经指出了我相信计算机的存在可能比其他新技术（包括电视甚至印刷）对智力发展产生更根本影响的一个原因。计算机作为数学实体的隐喻使学习者与重要的知识领域建立了一种全新的关系。即使是最好的教育电视节目也仅限于提供没有它时存在的学习类型的量化改进。“芝麻街”可能比孩子从一些父母或幼儿园老师那里得到的解释更好、更吸引人，但孩子仍然处于倾听解释的位置。相比之下，当孩子学习编程时，学习过程发生了变化。它变得更加积极和自我导向。特别是，知识是为了可识别的个人目的而获得的。孩子用它做一些事情。新知识是力量的源泉，从它在孩子的头脑中开始形成的那一刻起，孩子就体验到了这种力量。
I have already indicated one reason for my belief that the computer presence might have more fundamental effects on intellectual development than did other new technologies, including television and even printing. The metaphor of computer as mathematics-speaking entity puts the learner in a qualitatively new kind of relationship to an important domain of knowledge. Even the best of educational television is limited to offering quantitative improvements in the kinds of learning that existed without it. “Sesame Street” might offer better and more engaging explanations than a child can get from some parents or nursery school teachers, but the child is still in the position of listening to explanations. By contrast, when a child learns to program, the process of learning is transformed. It becomes more active and self-directed. In particular, the knowledge is acquired for a recognizable personal purpose. The child does something with it. The new knowledge is a source of power and is experienced as such from the moment it begins to form in the child’s mind.
我说过数学学习方式发生了变化。但影响的远不止数学。通过研究皮亚杰的另一个观点，我们可以了解变化的程度。皮亚杰区分了“具体”思维和“形式”思维。当孩子 6 岁进入一年级时，具体思维已经很好地形成了，并在接下来的几年里得到巩固。形式思维要到孩子几乎两倍大的时候才会发展，也就是说在 12 岁左右，前后一两年，一些研究人员甚至认为许多人从未实现完全的形式思维。我不完全接受皮亚杰的区分，但我确信它足够接近现实，可以帮助我们理解这样一种观点，即一项创新对智力发展的影响在质量上可能大于一千项其他创新的累积量效应。简单地说，我的猜想是计算机可以具体化（和个性化）形式。从这个角度来看，它不仅仅是另一个强大的教育工具。它的独特之处在于，它为我们提供了解决皮亚杰和其他许多人认为的从儿童思维到成人思维过程中需要克服的障碍的方法。我相信它可以让我们改变具体和形式之间的界限。过去只能通过形式过程获得的知识现在可以具体地获得。真正的魔力在于，这种知识包含了成为形式思考者所需的那些要素。
I have spoken of mathematics being learned in a new way. But much more is affected than mathematics. One can get an idea of the extent of what is changed by examining another of Piaget’s ideas. Piaget distinguishes between “concrete” thinking and “formal” thinking. Concrete thinking is already well on its way by the time the child enters the first grade at age 6 and is consolidated in the following several years. Formal thinking does not develop until the child is almost twice as old, that is to say at age 12, give or take a year or two, and some researchers have even suggested that many people never achieve fully formal thinking. I do not fully accept Piaget’s distinction, but I am sure that it is close enough to reality to help us make sense of the idea that the consequences for intellectual development of one innovation could be qualitatively greater than the cumulative quantitative effects of a thousand others. Stated most simply, my conjecture is that the computer can concretize (and personalize) the formal. Seen in this light, it is not just another powerful educational tool. It is unique in providing us with the means for addressing what Piaget and many others see as the obstacle which is overcome in the passage from child to adult thinking. I believe that it can allow us to shift the boundary separating concrete and formal. Knowledge that was accessible only through formal processes can now be approached concretely. And the real magic comes from the fact that this knowledge includes those elements one needs to become a formal thinker.
这种对计算机角色的描述相当抽象。我将通过研究使用计算机对皮亚杰与智力发展的形式阶段相关的两种思维的影响来具体化它：组合思维，即人们必须根据系统所有可能状态的集合进行推理，以及关于思维本身的自指思维。
This description of the role of the computer is rather abstract. I shall concretize it, anticipating discussions which occur in later chapters of this book, by looking at the effect of working with computers on two kinds of thinking Piaget associates with the formal stage of intellectual development: combinatorial thinking, where one has to reason in terms of the set of all possible states of a system, and self-referential thinking about thinking itself.
在一个典型的组合思维实验中，孩子们被要求组成各种颜色珠子的所有可能组合（或“系列”）。大多数孩子直到五六年级才能够系统而准确地完成这项工作，这确实非常了不起。为什么会这样？为什么这项任务似乎比七八岁儿童完成的智力壮举要困难得多？它的逻辑结构本质上更复杂吗？它可能需要一种直到青春期才成熟的神经机制吗？我认为，通过观察文化的性质可以得到更可能的解释。制作珠子系列的任务可以看作是构建和执行一个程序，这是一种非常常见的程序，其中嵌套了两个循环：确定第一个颜色并运行所有可能的第二个颜色，然后重复，直到运行完所有可能的第一个颜色。对于完全熟悉计算机和编程的人来说，这项任务没有任何“正式”或抽象之处。对于生活在计算机文化中的孩子来说，这就像在餐桌上搭配刀叉一样具体。即使是将某些家族包括两次（例如，红蓝和蓝红）的常见“错误”也是众所周知的。我们的文化中充满了各种各样的成对、成对和一一对应关系，并且有丰富的语言来谈论这些事情。这种丰富性既为孩子们提供了激励，也为孩子们提供了模型和工具，让他们建立思考诸如三块大糖果是否比四块小得多的糖果多或少等问题的方法。对于这类问题，我们的孩子获得了极好的数量直觉。但我们的文化在系统程序模型方面相对较差。直到最近，流行语言中甚至还没有编程的名称，更不用说成功编程所需的想法了。没有“嵌套循环”这个词，也没有重复计算错误的词。事实上，没有词来形容计算机专家所说的“错误”和“调试”等强大的想法。
In a typical experiment in combinatorial thinking, children are asked to form all the possible combinations (or “families”) of beads of assorted colors. It really is quite remarkable that most children are unable to do this systematically and accurately until they are in the fifth or sixth grades. Why should this be? Why does this task seem to be so much more difficult than the intellectual feats accomplished by seven- and eight-year-old children? Is its logical structure essentially more complex? Can it possibly require a neurological mechanism that does not mature until the approach of puberty? I think that a more likely explanation is provided by looking at the nature of the culture. The task of making the families of beads can be looked at as constructing and executing a program, a very common sort of program, in which two loops are nested: Fix a first color and run through all the possible second colors, then repeat until all possible first colors have been run through. For someone who is thoroughly used to computers and programming, there is nothing “formal” or abstract about this task. For a child in a computer culture, it would be as concrete as matching up knives and forks at the dinner table. Even the common “bug” of including some families twice (for example, red-blue and blue-red) would be well-known. Our culture is rich in pairs, couples, and one-to-one correspondences of all sorts, and it is rich in language for talking about such things. This richness provides both the incentive and a supply of models and tools for children to build ways to think about such issues as whether three large pieces of candy are more or less than four much smaller pieces. For such problems our children acquire an excellent intuitive sense of quantity. But our culture is relatively poor in models of systematic procedures. Until recently there was not even a name in popular language for programming, let alone for the ideas needed to do so successfully. There is no word for “nested loops” and no word for the double-counting bug. Indeed, there are no words for the powerful ideas computerists refer to as “bug” and “debugging.”
由于缺乏激励或材料来构建强有力的具体方法来思考涉及系统性的问题，孩子们被迫以摸索、抽象的方式处理这些问题。因此，美国城市和非洲村庄共有的文化因素可以解释孩子们在建立数量和系统性直觉知识的年龄差异。
Without the incentive or the materials to build powerful, concrete ways to think about problems involving systematicity, children are forced to approach such problems in a groping, abstract fashion. Thus cultural factors that are common to both the American city and the African village can explain the difference in age at which children build their intuitive knowledge of quantity and of systematicity.
当我还在日内瓦工作时，我就开始敏锐地意识到，当时非常年轻的计算机文化中的材料如何帮助心理学家开发出思考问题的新方法。1事实上，我进入计算机世界很大程度上是出于这样的想法：儿童也可以从计算机模型似乎能够将以前看起来如此无形和抽象的知识领域具体化的方式中受益，甚至可能比心理学家受益更多。
While still working in Geneva I had become sensitive to the way in which materials from the then very young computer cultures were allowing psychologists to develop new ways to think about thinking.1 In fact, my entry into the world of computers was motivated largely by the idea that children could also benefit, perhaps even more than the psychologists, from the way in which computer models seemed able to give concrete form to areas of knowledge that had previously appeared so intangible and abstract.
我开始明白，那些学会了编程的孩子如何利用非常具体的计算机模型来思考思考和学习学习，从而提高他们作为心理学家和认识论者的能力。例如，许多孩子在学习中受到阻碍，因为他们的学习模型要么“懂”要么“懂”。但是当你学习编程时，你几乎从来没有第一次就做对。学习成为一名高级程序员就是学习高超地隔离和纠正“错误”的能力，这些错误是阻碍程序运行的部分。关于程序的问题不是它是对是错，而是它是否可以修复。如果这种看待智力产品的方式推广到更广泛的文化对知识及其获取的看法，我们可能就不会那么害怕“犯错”。计算机对改变我们对成功和失败的黑白版本的看法的潜在影响就是将计算机用作“思考对象”的一个例子。显然，无需使用计算机即可获得良好的学习策略。当然，成功的学习者早在计算机出现之前就开发了“调试”策略。但是，将学习与开发程序进行类比是一种有效且可行的方法，可以开始更清楚地了解调试策略并更加刻意地改进它们。
I began to see how children who had learned to program computers could use very concrete computer models to think about thinking and to learn about learning and, in doing so, enhance their powers as psychologists and as epistemologists. For example, many children are held back in their learning because they have a model of learning in which you have either “got it” or “got it wrong.” But when you learn to program a computer, you almost never get it right the first time. Learning to be a master programmer is learning to become highly skilled at isolating and correcting “bugs,” the parts that keep the program from working. The question to ask about the program is not whether it is right or wrong, but if it is fixable. If this way of looking at intellectual products were generalized to how the larger culture thinks about knowledge and its acquisition, we all might be less intimidated by our fears of “being wrong.” This potential influence of the computer on changing our notion of a black and white version of our successes and failures is an example of using the computer as an “object-to-think-with.” It is obviously not necessary to work with computers in order to acquire good strategies for learning. Surely “debugging” strategies were developed by successful learners long before computers existed. But thinking about learning by analogy with developing a program is a powerful and accessible way to get started on becoming more articulate about one’s debugging strategies and more deliberate about improving them.
我对计算机文化及其对思维的影响的讨论，是以强大的计算机大规模渗透到人们的生活为前提的。毫无疑问，这种情况会发生。计算器、电子游戏和数字手表是由一场技术革命带给我们的，这场革命在其他所有电子产品价格都因通货膨胀而上涨的时期，迅速降低了电子产品的价格。集成电路带来的同一场技术革命，现在又给我们带来了个人电脑。大型计算机过去要花费数百万美元，因为它们是由数百万个物理上不同的部件组装而成的。在新技术中，复杂的电路不是组装而成的，而是作为一个整体、坚固的实体制造的——因此有“集成电路”一词。通过将集成电路技术与印刷术进行比较，可以理解它对成本的影响。制作一本书的主要开支发生在印刷机开始运转之前很久。它用于写作、编辑和排版。其他成本发生在印刷之后：装订、分发和营销。印刷本身的实际每本书的成本可以忽略不计。对于一本功能强大的书和一本平凡的书来说，情况都是如此。同样，集成电路的大部分成本都花在了准备过程中；只要有足够的销售量来分摊开发成本，制造单个电路的实际成本就可以忽略不计。这项技术对计算成本的影响是巨大的。20 世纪 60 年代要花费数十万美元，70 年代初要花费数万美元的计算机，现在只需不到一美元就能制造出来。唯一的限制因素是特定电路是否能装入对应于“页面”的东西中——也就是蚀刻电路的“硅片”上。
My discussion of a computer culture and its impact on thinking presupposes a massive penetration of powerful computers into people’s lives. That this will happen there can be no doubt. The calculator, the electronic game, and the digital watch were brought to us by a technical revolution that rapidly lowered prices for electronics in a period when all others were rising with inflation. That same technological revolution, brought about by the integrated circuit, is now bringing us the personal computer. Large computers used to cost millions of dollars because they were assembled out of millions of physically distinct parts. In the new technology a complex circuit is not assembled but made as a whole, solid entity—hence the term “integrated circuit.” The effect of integrated circuit technology on cost can be understood by comparing it to printing. The main expenditure in making a book occurs long before the press begins to roll. It goes into writing, editing, and typesetting. Other costs occur after the printing: binding, distributing, and marketing. The actual cost per copy for printing itself is negligible. And the same is true for a powerful as for a trivial book. So, too, most of the cost of an integrated circuit goes into a preparatory process; the actual cost of making an individual circuit becomes negligible, provided enough are sold to spread the costs of development. The consequences of this technology for the cost of computation are dramatic. Computers that would have cost hundreds of thousands in the 1960s and tens of thousands in the early 1970s can now be made for less than a dollar. The only limiting factor is whether the particular circuit can fit onto what corresponds to a “page”—that is to say the “silicon chips” on which the circuits are etched.
但是，每年，在硅片上蚀刻电路的技术都在以有规律和可预测的方式变得更加精湛。越来越复杂的电路可以压缩到芯片上，而以不到一美元的价格生产的计算机能力也在不断提高。我预测，在本世纪末之前，人们将购买具有与目前售价数百万美元的 IBM 计算机相当的计算机能力的儿童玩具。至于用于此类用途的计算机，这些机器的主要成本将是外围设备，例如键盘。即使这些设备的价格不下降，超级计算机的价格也可能与打字机和电视机的价格相当。
But each year in a regular and predictable fashion the art of etching circuits on silicon chips is becoming more refined. More and more complex circuitry can be squeezed onto a chip, and the computer power that can be produced for less than a dollar increases. I predict that long before the end of the century, people will buy children toys with as much computer power as the great IBM computers currently selling for millions of dollars. And as for computers to be used as such, the main cost of these machines will be the peripheral devices, such as the keyboard. Even if these do not fall in price, it is likely that a supercomputer will be equivalent in price to a typewriter and a television set.
专家们一致认为，计算机的成本将下降到一定水平，大量进入日常生活。有些计算机将以真正的计算机形式出现，即可编程机器。其他计算机则可能以越来越复杂的游戏形式出现，并出现在自动化超市中，货架，甚至罐头都会说话。人们真的可以尽情发挥自己的想象力。毫无疑问，每个人的生活物质层面都会变得截然不同，对孩子来说可能尤其如此。但对于计算机的存在将产生的影响，人们的看法存在很大分歧。我将我的思想与两种思想潮流区分开来，我在这里称之为“怀疑论”和“批判论”。
There really is no disagreement among experts that the cost of computers will fall to a level where they will enter everyday life in vast numbers. Some will be there as computers proper, that is to say, programmable machines. Others might appear as games of ever-increasing complexity and in automated supermarkets where the shelves, maybe even the cans, will talk. One really can afford to let one’s imagination run wild. There is no doubt that the material surface of life will become very different for everyone, perhaps most of all for children. But there has been significant difference of opinion about the effects this computer presence will produce. I would distinguish my thinking from two trends of thinking, which I refer to here as the “skeptical” and the “critical.”
怀疑论者并不认为计算机的存在会对人们的学习和思考方式产生多大影响。我已经提出了一些可能的解释来解释他们为什么会这样想。在某些情况下，我认为怀疑论者可能对教育及其计算机的影响的理解过于狭隘。他们没有考虑一般的文化影响，而是将注意力集中在将计算机用作程序化教学设备上。怀疑论者随后得出结论，尽管计算机可能会对学校学习产生一些改进，但它不太可能带来根本性的变化。从某种意义上说，我也认为怀疑论者的观点源于未能认识到皮亚杰式的学习在儿童成长过程中究竟发生了多少。如果一个人认为儿童的智力发展（或者道德或社会发展）主要源于刻意的教学，那么这个人很可能会低估大量计算机和其他交互式物体对儿童的潜在影响。
Skeptics do not expect the computer presence to make much difference in how people learn and think. I have formulated a number of possible explanations for why they think as they do. In some cases, I think the skeptics might conceive of education and the effect of computers on it too narrowly. Instead of considering general cultural effects, they focus attention on the use of the computer as a device for programmed instruction. Skeptics then conclude that while the computer might produce some improvements in school learning, it is not likely to lead to fundamental change. In a sense, too, I think the skeptical view derives from a failure to appreciate just how much Piagetian learning takes place as a child grows up. If a person conceives of children’s intellectual development (or, for that matter, moral or social development) as deriving chiefly from deliberate teaching, then such a person would be likely to underestimate the potential effect that a massive presence of computers and other interactive objects might have on children.
另一方面，批评者2认为计算机的存在将会带来影响，但他们也对此心存疑虑。例如，他们担心通过计算机进行更多的交流可能会导致人类交往减少，从而导致社会分裂。由于懂得如何使用计算机对于有效的社会和经济参与越来越必要，弱势群体的处境可能会恶化，计算机可能会加剧现有的阶级差异。至于计算机将产生的政治影响，批评者的担忧与奥威尔式的 1984 年的形象相呼应，当时家用计算机将成为复杂的监视和思想控制系统的一部分。批评者还提请人们注意计算机渗透可能带来的心理健康危害。其中一些危害是许多当代生活观察者已经担心的问题的放大形式；另一些则是本质上全新的问题。前一种类型的典型例子是，当我们考虑超级电视时代时，我们对电视心理影响的严重无知变得更加严重。计算机至少可以通过两种方式增强电视节目的吸引力和心理影响。内容可能会根据每个观众的口味而变化，节目可能会变得具有互动性，吸引“观众”参与其中。这些事情属于未来，但那些担心计算机对人的影响的人已经举出了学生彻夜不眠地盯着电脑终端，忽视学习和社交的例子。一些家长在看到自己的孩子对玩还很初级的电子游戏表现出特别的着迷时，想起了这些故事。
The critics,2 on the other hand, do think that the computer presence will make a difference and are apprehensive. For example, they fear that more communication via computers might lead to less human association and result in social fragmentation. As knowing how to use a computer becomes increasingly necessary to effective social and economic participation, the position of the underprivileged could worsen, and the computer could exacerbate existing class distinctions. As to the political effect computers will have, the critics’ concerns resonate with Orwellian images of a 1984 where home computers will form part of a complex system of surveillance and thought control. Critics also draw attention to potential mental health hazards of computer penetration. Some of these hazards are magnified forms of problems already worrying many observers of contemporary life; others are problems of an essentially new kind. A typical example of the former kind is that our grave ignorance of the psychological impact of television becomes even more serious when we contemplate an epoch of super TV. The holding power and the psychological impact of the television show could be increased by the computer in at least two ways. The content might be varied to suit the tastes of each individual viewer, and the show might become interactive, drawing the “viewer” into the action. Such things belong to the future, but people who are worried about the impact of the computer on people already cite cases of students spending sleepless nights riveted to the computer terminal, coming to neglect both studies and social contact. Some parents have been reminded of these stories when they observe a special quality of fascination in their own children’s reaction to playing with the still rudimentary electronic games.
在新问题而非旧问题的加剧方面，批评者指出，计算机的机械化思维过程对人类思维方式的影响。马歇尔·麦克卢汉的名言“媒介即信息”可能适用于此：如果媒介是一个交互系统，它接收单词并像人一样回复，那么很容易得到这样的信息：机器就像人，人就像机器。这对成长中的孩子的价值观和自我形象的发展可能产生什么影响很难评估。但不难看出担忧的理由。
In the category of problems that are new rather than aggravated versions of old ones, critics have pointed to the influence of the allegedly mechanized thought processes of computers on how people think. Marshall McCluhan’s dictum that “the medium is the message” might apply here: If the medium is an interactive system that takes in words and speaks back like a person, it is easy to get the message that machines are like people and that people are like machines. What this might do to the development of values and self-image in growing children is hard to assess. But it is not hard to see reasons for worry.
尽管有这些担忧，但我对计算机对社会的影响本质上是乐观的——有些人可能会说是乌托邦式的。我不会驳斥批评者的观点。相反，我也认为计算机的存在对人类思维具有强大的影响。我非常清楚交互式计算机的吸引力，以及以计算机为模型可以如何影响我们对自己的看法。事实上，我在过去十年中投入了大量精力进行LOGO研究，正是在积极方向上发展这种力量。例如，批评者一想到一个孩子被一台未来派的、计算机化的超级弹球机催眠，就会感到震惊。在LOGO研究中，我们发明了这种机器的版本，其中嵌入了物理学、数学或语言学的强大思想，使玩家能够以自然的方式学习它们，类似于孩子学习说话的方式。批评者如此担心的计算机“吸引力”成为一种有用的教育工具。或者再举一个更深刻的例子。评论家担心孩子们会以计算机为榜样，最终自己也会“机械地思考”。我采取了相反的策略，发明了一些方法，利用教育机会掌握像计算机一样刻意思考的艺术，例如，根据计算机程序的刻板印象，计算机程序以循序渐进、字面化、机械的方式进行。在某些情况下，这种思维方式是合适和有用的。一些孩子在学习语法或数学等正式科目时遇到的困难源于他们无法理解这种思维方式的意义。
Despite these concerns I am essentially optimistic—some might say utopian—about the effect of computers on society. I do not dismiss the arguments of the critics. On the contrary, I too see the computer presence as a potent influence on the human mind. I am very much aware of the holding power of an interactive computer and of how taking the computer as a model can influence the way we think about ourselves. In fact the work on LOGO to which I have devoted much of the past ten years consists precisely of developing such forces in positive directions. For example, the critic is horrified at the thought of a child hypnotically held by a futuristic, computerized super-pinball machine. In the LOGO work we have invented versions of such machines in which powerful ideas from physics or mathematics or linguistics are embedded in a way that permits the player to learn them in a natural fashion, analogous to how a child learns to speak. The computer’s “holding power,” so feared by critics, becomes a useful educational tool. Or take another, more profound example. The critic is afraid that children will adopt the computer as model and eventually come to “think mechanically” themselves. Following the opposite tack, I have invented ways to take educational advantage of the opportunities to master the art of deliberately thinking like a computer, according, for example, to the stereotype of a computer program that proceeds in a step-by-step, literal, mechanical fashion. There are situations where this style of thinking is appropriate and useful. Some children’s difficulties in learning formal subjects such as grammar or mathematics derive from their inability to see the point of such a style.
第二个教育优势是间接的，但最终更为重要。通过刻意学习模仿机械思维，学习者能够清楚地表达什么是机械思维，什么不是机械思维。这种练习可以增强选择适合问题的认知方式的能力。分析“机械思维”及其与其他类型思维的不同之处，并练习问题分析，可以提高智力水平。通过提供特定思维方式的非常具体、实用的模型，使用计算机可以更容易地理解“思维方式”的存在。让孩子们有机会选择一种或另一种方式，为他们提供了发展在两种方式之间进行选择所需的技能的机会。因此，与计算机接触可能不是诱导机械思维，而是可以想象到的最好的解药。对我来说，最重要的是，通过这些经历，这些孩子将作为认识论者进行学徒，也就是说，学习清晰地思考思维。
A second educational advantage is indirect but ultimately more important. By deliberately learning to imitate mechanical thinking, the learner becomes able to articulate what mechanical thinking is and what it is not. The exercise can lead to greater confidence about the ability to choose a cognitive style that suits the problem. Analysis of “mechanical thinking” and how it is different from other kinds and practice with problem analysis can result in a new degree of intellectual sophistication. By providing a very concrete, down-to-earth model of a particular style of thinking, work with the computer can make it easier to understand that there is such a thing as a “style of thinking.” And giving children the opportunity to choose one style or another provides an opportunity to develop the skill necessary to choose between styles. Thus instead of inducing mechanical thinking, contact with computers could turn out to be the best conceivable antidote to it. And for me what is most important in this is that through these experiences these children would be serving their apprenticeships as epistemologists, that is to say learning to think articulately about thinking.
当今文化为儿童提供的智力环境缺乏机会将他们对思考的思考公开化、学会谈论思考并通过外化来测试他们的想法。使用计算机可以极大地改变这种情况。即使是最简单的海龟工作也可以为磨练一个人对思考的思考提供新的机会：对海龟进行编程首先要让人反思自己如何做自己希望海龟做的事情。因此，教海龟行动或“思考”可以引导人们反思自己的行为和思想。随着孩子们的进步，他们会对计算机进行编程以做出更复杂的决策，并发现自己正在反思自己思维的更复杂方面。
The intellectual environments offered to children by today’s cultures are poor in opportunities to bring their thinking about thinking into the open, to learn to talk about it, and to test their ideas by externalizing them. Access to computers can dramatically change this situation. Even the simplest Turtle work can open new opportunities for sharpening one’s thinking about thinking: Programming the Turtle starts by making one reflect on how one does oneself what one would like the Turtle to do. Thus teaching the Turtle to act or to “think” can lead one to reflect on one’s own actions and thinking. And as children move on, they program the computer to make more complex decisions and find themselves engaged in reflecting on more complex aspects of their own thinking.
简而言之，虽然评论家和我都相信使用计算机可以对人们的思维方式产生强大的影响，但我将注意力转向探索如何将这种影响转化为积极的方向。
In short, while the critic and I share the belief that working with computers can have a powerful influence on how people think, I have turned my attention to exploring how this influence could be turned in positive directions.
我看到两种反对我反对批评者观点的反驳。第一种反对我的观点，即儿童成为认识论者是一件好事。许多人会认为，过度分析、言语化的思考是适得其反的，即使是故意选择的。第二种反对意见反对我的观点，即计算机可能会导致更具反思性的自我意识思考。许多人会认为，使用计算机通常会产生相反的效果。这两种反对意见需要不同类型的分析，不能同时讨论。第一种反对意见提出了关于学习心理学的技术问题，这将在第 4 章和第 6 章中讨论。第二种反对意见最直接的回答是，计算机绝对不会产生我希望看到的效果。并非所有计算机系统都如此。当今使用的大多数计算机系统都不是。在LOGO环境中，我看到孩子们在试图将自己的个人知识捕捉到一个程序中，让海龟执行他们自己非常了解如何执行的操作时，热烈地谈论他们自己的个人知识。但当然，计算机的物理存在并不足以确保此类对话的发生。远非如此。在数千所学校和数万个私人家庭中，孩子们现在正在经历着截然不同的计算机体验。在大多数情况下，计算机要么被用作多功能视频游戏，要么被用作“教学机器”，用于让孩子们练习算术或拼写。即使当父母、同龄人或专业教师教孩子用BASIC等语言编写简单程序时，这种活动也完全没有伴随我们在LOGO环境中看到的那种认识论反思。因此，我与批评者一样对现在计算的方式持怀疑态度。但我感兴趣的是刺激事物发展的重大变化。这种变化的底线是政治性的。现在发生的事情是一个经验问题。可能发生的情况是一个技术问题。但将会发生什么是一个政治问题，取决于社会选择。
I see two kinds of counterarguments to my arguments against the critics. The first kind challenges my belief that it is a good thing for children to be epistemologists. Many people will argue that overly analytic, verbalized thinking is counterproductive, even if it is deliberately chosen. The second kind of objection challenges my suggestion that computers are likely to lead to more reflective self-conscious thinking. Many people will argue that work with computers usually has the opposite effect. These two kinds of objections call for different kinds of analysis and cannot be discussed simultaneously. The first kind raises technical questions about the psychology of learning, which will be discussed in chapters 4 and 6. The second kind of objection is most directly answered by saying that there is absolutely no inevitability that computers will have the effects I hope to see. Not all computer systems do. Most in use today do not. In LOGO environments I have seen children engaged in animated conversations about their own personal knowledge as they try to capture it in a program to make a Turtle carry out an action that they themselves know very well how to do. But of course the physical presence of a computer is not enough to ensure that such conversations will come about. Far from it. In thousands of schools and in tens of thousands of private homes children are right now living through very different computer experiences. In most cases the computer is being used either as a versatile video game or as a “teaching machine” programmed to put children through their paces in arithmetic or spelling. And even when children are taught by a parent, a peer, or a professional teacher to write simple programs in a language like BASIC, this activity is not accompanied at all by the kind of epistemological reflection that we see in the LOGO environments. So I share a skepticism with the critics about what is being done with computation now. But I am interested in stimulating a major change in how things can be. The bottom line for such changes is political. What is happening now is an empirical question. What can happen is a technical question. But what will happen is a political question, depending on social choices.
关于 20 世纪 80 年代计算机对儿童的影响，主要未解决的问题是：哪些人会被计算机世界所吸引，他们会带来哪些才能，他们会给日益发展的计算机文化带来什么样的品味和意识形态？我曾描述过孩子们在LOGO环境中进行关于他们自己思考的自我参照讨论。之所以会发生这种情况，是因为LOGO语言和 Turtle 是由喜欢这种讨论并努力设计一种鼓励这种讨论的媒介的人设计的。其他计算机系统设计者对哪些类型的活动适合儿童有不同的品味和不同的想法。哪种设计会占上风，在哪种亚文化中占上风，不会由简单的官僚决策（例如政府教育部或专家委员会）决定。计算机风格的趋势将在一个复杂的决策网络中出现，这些决策来自拥有资源支持某一种设计的基金会、可能看到市场的公司、学校、决定在新的活动领域发展事业的个人，以及对选择和制作内容有自己发言权的儿童。人们常常问，未来的孩子是会编程计算机还是会专注于预先编程的活动。答案一定是，有些孩子会做前者，有些会做后者，有些两者兼而有之，有些两者都不做。但哪些孩子，最重要的是，哪些社会阶层的孩子，会属于哪一类，将受到计算机活动的类型和他们周围所创造的环境的影响。
The central open questions about the effect of computers on children in the 1980s are these: Which people will be attracted to the world of computers, what talents will they bring, and what tastes and ideologies will they impose on the growing computer culture? I have described children in LOGO environments engaged in self-referential discussions about their own thinking. This could happen because the LOGO language and the Turtle were designed by people who enjoy such discussion and worked hard to design a medium that would encourage it. Other designers of computer systems have different tastes and different ideas about what kinds of activities are suitable for children. Which design will prevail, and in what subculture, will not be decided by a simple bureaucratic decision made, for example, in a government Department of Education or by a committee of experts. Trends in computer style will emerge from a complex web of decisions by foundations with resources to support one or another design, by corporations who may see a market, by schools, by individuals who will decide to make their career in the new field of activity, and by children who will have their own say in what they pick up and what they make of it. People often ask whether in the future children will program computers or become absorbed in pre-programmed activities. The answer must be that some children will do the one, some the other, some both, and some neither. But which children, and most importantly, which social classes of children, will fall into each category will be influenced by the kind of computer activities and the kind of environments created around them.
举个例子，我们来考虑一个大多数人在想到计算机和儿童时可能不会想到的活动：使用计算机作为书写工具。对我来说，写作意味着起草草稿，并在相当长的一段时间内对其进行改进。我对自己作为作家的形象包括对“不可接受的”初稿的期望，该初稿将通过连续的编辑发展为可呈现的形式。但如果我是三年级学生，我就无法承受这种形象。写作的体力活动会很慢而且很费力。我没有秘书。对于大多数孩子来说，重写文本非常费力，以至于初稿就是最终稿，他们永远无法掌握以批判的眼光重读的技能。当孩子们接触到能够处理文本的计算机时，这种情况发生了巨大变化。初稿是在键盘上撰写的。很容易进行修改。当前的副本总是干净整洁。我看到一个孩子在开始用电脑写作的几周内从完全拒绝写作转变为强烈参与（伴随着质量的快速提高）。当孩子有身体残疾，手写比平时更加困难甚至不可能时，就会出现更为显著的变化。
As an example, we consider an activity which may not occur to most people when they think of computers and children: the use of a computer as a writing instrument. For me, writing means making a rough draft and refining it over a considerable period of time. My image of myself as a writer includes the expectation of an “unacceptable” first draft that will develop with successive editing into presentable form. But I would not be able to afford this image if I were a third grader. The physical act of writing would be slow and laborious. I would have no secretary. For most children rewriting a text is so laborious that the first draft is the final copy, and the skill of rereading with a critical eye is never acquired. This changes dramatically when children have access to computers capable of manipulating text. The first draft is composed at the keyboard. Corrections are made easily. The current copy is always neat and tidy. I have seen a child move from total rejection of writing to an intense involvement (accompanied by rapid improvement of quality) within a few weeks of beginning to write with a computer. Even more dramatic changes are seen when the child has physical handicaps that make writing by hand more than usually difficult or even impossible.
任何以写作为生的成年人都迅速采用了这种使用电脑的方法。现在，大多数报纸都为员工配备了“文字处理”电脑系统。许多在家工作的作家都购买了自己的电脑，电脑终端正逐渐取代打字机成为秘书的基本工具。孩子们使用电脑作为写作工具的形象很好地证明了我的一般论点：对专业人士有益的东西对孩子也有益。但是，这种关于电脑如何有助于儿童掌握语言的形象与大多数小学中根深蒂固的形象截然相反。在小学里，电脑被视为一种教学工具。它让孩子们练习区分动词和名词、拼写以及回答关于文本含义的多项选择题。在我看来，这种差异并不是两种教学策略之间微小的技术选择问题。它反映了教育理念的根本差异。更重要的是，它反映了对童年本质的不同看法。我认为，计算机作为书写工具，为儿童提供了一个机会，使他们在与智力产品和自我的关系上变得更像成年人，甚至更像高级专业人士。这样一来，它就与学校的许多方面发生了正面冲突，而学校的效果，如果不是其意图的话，就是让孩子“幼稚化”。
This use of computers is rapidly becoming adopted wherever adults write for a living. Most newspapers now provide their staff with “word processing” computer systems. Many writers who work at home are acquiring their own computers, and the computer terminal is steadily displacing the typewriter as the secretary’s basic tool. The image of children using the computer as a writing instrument is a particularly good example of my general thesis that what is good for professionals is good for children. But this image of how the computer might contribute to children’s mastery of language is dramatically opposed to the one that is taking root in most elementary schools. There the computer is seen as a teaching instrument. It gives children practice in distinguishing between verbs and nouns, in spelling, and in answering multiple-choice questions about the meaning of pieces of text. As I see it, this difference is not a matter of a small and technical choice between two teaching strategies. It reflects a fundamental difference in educational philosophies. More to the point, it reflects a difference in views on the nature of childhood. I believe that the computer as writing instrument offers children an opportunity to become more like adults, indeed like advanced professionals, in their relationship to their intellectual products and to themselves. In doing so, it comes into head-on collision with the many aspects of school whose effect, if not whose intention, is to “infantilize” the child.
文字处理器可以让孩子的写作体验更像真正的作家。但如果孩子周围的成年人不懂得作家的感受，这种体验就会受到破坏。例如，很容易想象成年人（包括教师）会表达这样的观点：编辑和重新编辑文本是在浪费时间（“你为什么不开始写点新东西？”或“你写得一点儿也不好，为什么不改正你的拼写？”）。
Word processors can make a child’s experience of writing more like that of a real writer. But this can be undermined if the adults surrounding that child fail to appreciate what it is like to be a writer. For example, it is only too easy to imagine adults, including teachers, expressing the view that editing and re-editing a text is a waste of time (“Why don’t you get on to something new?” or “You aren’t making it any better, why don’t you fix your spelling?”).
写作、音乐创作、技巧游戏、复杂图形等都是如此：计算机本身不是一种文化，但它可以促进截然不同的文化和哲学观点。例如，人们可以将海龟视为一种教授传统课程元素（如角度、形状和坐标系的概念）的设备。事实上，大多数向我咨询其用途的教师都试图以这种方式使用它，这完全可以理解。他们的问题是关于课堂组织、日程安排问题、海龟的引入所引起的教学问题，尤其是它与其他课程在概念上的关系。当然，海龟可以帮助教授传统课程，但我认为它是皮亚杰学习的载体，对我来说，这是没有课程的学习。
As with writing, so with music-making, games of skill, complex graphics, whatever: The computer is not a culture unto itself, but it can serve to advance very different cultural and philosophical outlooks. For example, one could think of the Turtle as a device to teach elements of the traditional curriculum, such as notions of angle, shape, and coordinate systems. And in fact, most teachers who consult me about its use are, quite understandably, trying to use it in this way. Their questions are about classroom organization, scheduling problems, pedagogical issues raised by the Turtle’s introduction, and especially, about how it relates conceptually to the rest of the curriculum. Of course the Turtle can help in the teaching of traditional curriculum, but I have thought of it as a vehicle for Piagetian learning, which to me is learning without curriculum.
有些人想创建“皮亚杰课程”或“皮亚杰教学方法”。但在我看来，这些短语和它们所代表的活动是自相矛盾的。我认为皮亚杰是无课程学习的理论家，也是无意识教学学习的理论家。把他变成新课程的理论家就是颠倒他的观点。
There are those who think about creating a “Piagetian curriculum” or “Piagetian teaching methods.” But to my mind these phrases and the activities they represent are contradictions in terms. I see Piaget as the theorist of learning without curriculum and the theorist of the kind of learning that happens without deliberate teaching. To turn him into the theorist of a new curriculum is to stand him on his head.
但“无课程教学”并不意味着自发、自由形式的课堂，也不意味着简单地“让孩子独自一人”。它意味着支持孩子利用周围文化中的材料构建自己的知识结构。在这种模式下，教育干预意味着改变文化，在其中植入新的建设性元素，并消除有害元素。这是一项比引入课程改革更雄心勃勃的任务，但在目前出现的条件下是可行的。
But “teaching without curriculum” does not mean spontaneous, free-form classrooms or simply “leaving the child alone.” It means supporting children as they build their own intellectual structures with materials drawn from the surrounding culture. In this model, educational intervention means changing the culture, planting new constructive elements in it, and eliminating noxious ones. This is a more ambitious undertaking than introducing a curriculum change, but one which is feasible under conditions now emerging.
假设三十年前，一位教育家决定，解决数学教育问题的方法是安排相当一部分人口熟练掌握（并热衷于）一种新的数学语言。这个想法在原则上可能很好，但在实践中却很荒谬。没有人有能力实现它。现在情况不同了。数百万人学习编程语言的原因与儿童教育无关。因此，影响他们学习语言的形式以及他们的孩子学习这些语言的可能性成为一个切实可行的建议。
Suppose that thirty years ago an educator had decided that the way to solve the problem of mathematics education was to arrange for a significant fraction of the population to become fluent in (and enthusiastic about) a new mathematical language. The idea might have been good in principle, but in practice it would have been absurd. No one had the power to implement it. Now things are different. Many millions of people are learning programming languages for reasons that have nothing to do with the education of children. Therefore, it becomes a practical proposition to influence the form of the languages they learn and the likelihood that their children will pick up these languages.
教育者必须是人类学家。作为人类学家的教育者必须努力了解哪些文化材料与智力发展相关。然后，他或她需要了解文化中正在发生哪些趋势。有意义的干预必须采取顺应这些趋势的形式。在我作为人类学家的教育者的角色中，我看到计算机渗透到个人生活中产生了新的需求。家里有电脑或在工作中使用电脑的人会希望能够和他们的孩子谈论电脑。他们希望能够教他们的孩子使用这些机器。因此，对海龟图之类的东西可能会有文化需求，而对新数学则从未有过、也许永远不会有文化需求。
The educator must be an anthropologist. The educator as anthropologist must work to understand which cultural materials are relevant to intellectual development. Then, he or she needs to understand which trends are taking place in the culture. Meaningful intervention must take the form of working with these trends. In my role of educator as anthropologist I see new needs being generated by the penetration of the computer into personal lives. People who have computers at home or who use them at work will want to be able to talk about them to their children. They will want to be able to teach their children to use the machines. Thus there could be a cultural demand for something like Turtle graphics in a way there never was, and perhaps never could be, a cultural demand for the New Math.
在本章中，我一直在谈论教育工作者、基金会、政府和个人的选择如何影响儿童学习方式的潜在革命性变化。但做出正确的选择并不总是容易的，部分原因是过去的选择常常会困扰我们。新技术的第一个可用但仍然原始的产品往往会陷入困境。我把这种现象称为QWERTY现象。
Throughout the course of this chapter I have been talking about the ways in which choices made by educators, foundations, governments, and private individuals can affect the potentially revolutionary changes in how children learn. But making good choices is not always easy, in part because past choices can often haunt us. There is a tendency for the first usable, but still primitive, product of a new technology to dig itself in. I have called this phenomenon the QWERTY phenomenon.
标准打字机最上面一排字母键是QWERTY。对我来说，这象征着技术往往不是推动进步的力量，而是让事物停滞不前。QWERTY排列方式没有合理的解释，只有历史原因。它是为了解决打字机早期的一个问题而引入的：按键容易卡住。其想法是通过分开经常连续的按键来尽量减少碰撞问题。仅仅几年后，技术的普遍改进解决了卡住问题，但QWERTY却保留了下来。它一经采用，就产生了数百万台打字机和一种学习打字的方法（实际上是一门成熟的课程）。变革的社会成本（例如，将键盘上最常用的按键放在一起）随着如此多的手指知道如何按照QWERTY键盘打字而产生的既得利益而不断增加。尽管存在其他更“合理”的系统，但QWERTY仍然保留了下来。另一方面，如果你和人们谈论QWERTY 键盘的排列方式，他们会用“客观”标准来证明其合理性。他们会告诉你，它“优化了这一点”或“最小化了那一点”。尽管这些理由没有合理的基础，但它们说明了一种神话构建过程，一种社会过程，使我们能够为任何系统的原始性找到理由。我认为，我们在计算机领域也正在走上同样的道路。我们正处于一种时代错误中，因为我们保留了那些除了技术和理论发展早期的历史根源之外没有任何合理基础的做法。
The top row of alphabetic keys of the standard typewriter reads QWERTY. For me this symbolizes the way in which technology can all too often serve not as a force for progress but for keeping things stuck. The QWERTY arrangement has no rational explanation, only a historical one. It was introduced in response to a problem in the early days of the typewriter: The keys used to jam. The idea was to minimize the collision problem by separating those keys that followed one another frequently. Just a few years later, general improvements in the technology removed the jamming problem, but QWERTY stuck. Once adopted, it resulted in many millions of typewriters and a method (indeed a full-blown curriculum) for learning typing. The social cost of change (for example, putting the most used keys together on the keyboard) mounted with the vested interest created by the fact that so many fingers now knew how to follow the QWERTY keyboard. QWERTY has stayed on despite the existence of other, more “rational” systems. On the other hand, if you talk to people about the QWERTY arrangement, they will justify it by “objective” criteria. They will tell you that it “optimizes this” or it “minimizes that.” Although these justifications have no rational foundation, they illustrate a process, a social process, of myth construction that allows us to build a justification for primitivity into any system. And I think that we are well on the road to doing exactly the same thing with the computer. We are in the process of digging ourselves into an anachronism by preserving practices that have no rational basis beyond their historical roots in an earlier period of technological and theoretical development.
使用计算机进行练习和实践只是计算机领域QWERTY现象的一个例子。另一个例子甚至发生在尝试让学生学习计算机编程时。正如我们将在后面的章节中看到的那样，学习计算机编程需要学习一种“编程语言”。有很多这样的语言——例如，FORTRAN、PASCAL、BASIC、SMALLTALK和LISP，以及鲜为人知的LOGO语言，我们的小组在大多数计算机和儿童实验中都使用了这种语言。当我们选择让孩子们学习计算机编程的语言时，可以预料到会出现强大的QWERTY现象。我将详细论证这个问题的重要性。编程语言就像一种自然的人类语言，它偏爱某些隐喻、图像和思维方式。所使用的语言强烈地影响着计算机文化。似乎对使用计算机感兴趣并对文化影响敏感的教育者会特别关注语言的选择。但事实并非如此。相反，教育工作者在技术问题上过于胆怯或太无知，无法试图影响计算机制造商提供的语言，他们接受某些编程语言的方式与他们接受QWERTY键盘的方式非常相似。一个有启发性的例子是编程语言BASIC 3已成为教授美国儿童如何编程计算机的首选语言。相关的技术信息是：一台非常小的计算机可以理解BASIC，而其他语言对计算机的要求更高。因此，在计算机能力极其昂贵的早期，使用BASIC有一个真正的技术原因，特别是在预算总是紧张的学校。今天，事实上，几年来，计算机内存的成本已经下降到使用BASIC的任何剩余经济优势都微不足道的地步。然而，在大多数高中，这种语言仍然几乎是编程的代名词，尽管存在其他明显更容易学习的计算机语言，并且学习它们可以带来更丰富的智力收益。这种情况自相矛盾。计算机革命才刚刚开始，就已经滋生了保守主义。仔细观察BASIC可以让我们了解保守的社会制度如何挪用并试图压制一种具有革命性潜力的工具。
The use of computers for drill and practice is only one example of the QWERTY phenomenon in the computer domain. Another example occurs even when attempts are made to allow students to learn to program the computer. As we shall see in later chapters, learning to program a computer involves learning a “programming language.” There are many such languages—for example, FORTRAN, PASCAL, BASIC, SMALLTALK, and LISP, and the lesser known language LOGO, which our group has used in most of our experiments with computers and children. A powerful QWERTY phenomenon is to be expected when we choose the language in which children are to learn to program computers. I shall argue in detail that the issue is consequential. A programming language is like a natural, human language in that it favors certain metaphors, images, and ways of thinking. The language used strongly colors the computer culture. It would seem to follow that educators interested in using computers and sensitive to cultural influences would pay particular attention to the choice of language. But nothing of the sort has happened. On the contrary, educators, too timid in technological matters or too ignorant to attempt to influence the languages offered by computer manufacturers, have accepted certain programming languages in much the same way as they accepted the QWERTY keyboard. An informative example is the way in which the programming language BASIC3 has established itself as the obvious language to use in teaching American children how to program computers. The relevant technical information is this: A very small computer can be made to understand BASIC, while other languages demand more from the computer. Thus, in the early days when computer power was extremely expensive, there was a genuine technical reason for the use of BASIC, particularly in schools where budgets were always tight. Today, and in fact for several years now, the cost of computer memory has fallen to the point where any remaining economic advantages of using BASIC are insignificant. Yet in most high schools, the language remains almost synonymous with programming, despite the existence of other computer languages that are demonstrably easier to learn and are richer in the intellectual benefits that can come from learning them. The situation is paradoxical. The computer revolution has scarcely begun but is already breeding its own conservatism. Looking more closely at BASIC provides a window on how a conservative social system appropriates and tries to neutralize a potentially revolutionary instrument.
BASIC之于计算，就如同QWERTY之于打字。许多教师都学过BASIC，许多书籍都写过关于 BASIC 的内容，许多计算机都以BASIC为“硬连线”的方式构建。在打字机的例子中，我们注意到人们如何发明“合理化”来为现状辩护。在BASIC的例子中，这种现象已经发展到类似于意识形态形成的地步。人们发明了复杂的论据来为BASIC的特性辩护，这些特性最初被纳入是因为原始技术需要它们，或者因为在设计语言时替代方案还不够为人所知。
BASIC is to computation as QWERTY is to typing. Many teachers have learned BASIC, many books have been written about it, many computers have been built in such a way that BASIC is “hardwired” into them. In the case of the typewriter, we noted how people invent “rationalizations” to justify the status quo. In the case of BASIC, the phenomenon has gone much further, to the point where it resembles ideology formation. Complex arguments are invented to justify features of BASIC that were originally included because the primitive technology demanded them or because alternatives were not well enough known at the time the language was designed.
BASIC意识形态的一个例子是，有人认为BASIC易于学习，因为它的词汇量非常小。如果我们将这一论点应用到儿童学习自然语言的背景下，这一论点的表面有效性就会立即受到质疑。想象一下，有人建议我们发明一种特殊的语言来帮助孩子们学习说话。这种语言的词汇量很小，只有 50 个词，但这 50 个词的选择非常巧妙，可以用它们表达所有的想法。这种语言会更容易学习吗？也许这种词汇很容易学，但用这些词汇来表达一个人想说的话会非常复杂，只有最有上进心和最聪明的孩子才会学会说“嗨”以外的词。这与BASIC的情况很接近。它的词汇量很小，可以很快学会。但使用它又是另一回事。BASIC 中的程序具有如此复杂的结构，以至于实际上只有最有上进心和最聪明（“数学”）的孩子才会学会用它来完成更多琐碎的事情。
An example of BASIC ideology is the argument that BASIC is easy to learn because it has a very small vocabulary. The surface validity of the argument is immediately called into question if we apply it to the context of how children learn natural languages. Imagine a suggestion that we invent a special language to help children learn to speak. This language would have a small vocabulary of just fifty words, but fifty words so well chosen that all ideas could be expressed using them. Would this language be easier to learn? Perhaps the vocabulary might be easy to learn, but the use of the vocabulary to express what one wanted to say would be so contorted that only the most motivated and brilliant children would learn to say more than “hi.” This is close to the situation with BASIC. Its small vocabulary can be learned quickly enough. But using it is a different matter. Programs in BASIC acquire so labyrinthine a structure that in fact only the most motivated and brilliant (“mathematical”) children do learn to use it for more than trivial ends.
有人可能会问，为什么老师们没有注意到孩子们学习BASIC 的困难。答案很简单：大多数老师并不期望大多数学生有很高的表现，尤其是在编程这种看似“数学化”和“形式化”的工作领域。因此，文化普遍认为数学难以理解，这加强了BASIC的维护，而这反过来又证实了这些看法。此外，教师并不是唯一将假设和偏见输入到延续BASIC 的循环中的人。还有计算机专家，他们是计算机世界中决定他们的计算机将使用哪种语言的人。这些人通常是工程师，他们发现BASIC非常容易学习，部分原因是他们习惯于学习这种非常技术性的系统，部分原因是BASIC的简单性符合他们的价值体系。因此，一种由计算机工程师主导的特定亚文化正在影响教育界，使其青睐那些最像这种亚文化的学生。这个过程是心照不宣的、无意的：它从未被公开表达过，更不用说评估了。从所有这些方面来看，BASIC的社会嵌入比QWERTY的“深入”有着更为严重的后果。
One might ask why the teachers do not notice the difficulty children have in learning BASIC. The answer is simple: Most teachers do not expect high performance from most students, especially in a domain of work that appears to be as “mathematical” and “formal” as programming. Thus the culture’s general perception of mathematics as inaccessible bolsters the maintenance of BASIC, which in turn confirms these perceptions. Moreover, the teachers are not the only people whose assumptions and prejudices feed into the circuit that perpetuates BASIC. There are also the computerists, the people in the computer world who make decisions about what languages their computers will speak. These people, generally engineers, find BASIC quite easy to learn, partly because they are accustomed to learning such very technical systems and partly because BASIC’s sort of simplicity appeals to their system of values. Thus, a particular subculture, one dominated by computer engineers, is influencing the world of education to favor those school students who are most like that subculture. The process is tacit, unintentional: It has never been publicly articulated, let alone evaluated. In all of these ways, the social embedding of BASIC has far more serious consequences than the “digging in” of QWERTY.
计算机相关的亚文化的属性还以许多其他方式投射到教育界。例如，计算机作为一种训练和实践工具的想法对教师很有吸引力，因为它类似于传统的教学方法，对设计计算机系统的工程师也有吸引力：训练和实践应用是可预测的、易于描述、高效利用机器资源。因此，最优秀的工程人才会投入到偏向于此类应用的计算机系统的开发中。这种偏见是微妙的。机器设计师实际上并不决定在课堂上做什么。那是由教师决定的，有时甚至由精心控制的比较研究实验决定。但这些受控实验有一个讽刺之处。他们非常擅长判断最佳成绩中看到的微小影响是真实的还是偶然的。但他们无法衡量机器内置偏见无疑真实（可能更严重）的影响。
There are many other ways in which the attributes of the subcultures involved with computers are being projected onto the world of education. For example, the idea of the computer as an instrument for drill and practice that appeals to teachers because it resembles traditional teaching methods also appeals to the engineers who design computer systems: Drill and practice applications are predictable, simple to describe, efficient in use of the machine’s resources. So the best engineering talent goes into the development of computer systems that are biased to favor this kind of application. The bias operates subtly. The machine designers do not actually decide what will be done in the classrooms. That is done by teachers and occasionally even by carefully controlled comparative research experiments. But there is an irony in these controlled experiments. They are very good at telling whether the small effects seen in best scores are real or due to chance. But they have no way to measure the undoubtedly real (and probably more massive) effects of the biases built into the machines.
我们已经注意到，在教育中使用计算机时所固有的保守偏见也已渗透到其他新技术中。新技术的首次使用很自然地会以略有不同的方式完成以前没有使用过它的工作。汽车设计师花了数年时间才接受了汽车而不是“无马马车”的概念，现代电影的前身是戏剧，好像是在现场观众面前表演，但实际上是在镜头前表演。整整一代人的时间才让电影这种新艺术成为一种与戏剧和摄影的线性混合截然不同的东西。迄今为止，以“教育技术”或“计算机在教育中的应用”为名所做的大部分工作仍处于旧教学方法与新技术的线性混合阶段。我将要讨论的主题是一些初步探索，旨在将基本教育原则与将其转化为现实的新方法进行更有机的互动。
We have already noted that the conservative bias being built into the use of computers in education has also been built into other new technologies. The first use of the new technology is quite naturally to do in a slightly different way what had been done before without it. It took years before designers of automobiles accepted the idea that they were cars, not “horseless carriages,” and the precursors of modern motion pictures were plays acted as if before a live audience but actually in front of a camera. A whole generation was needed for the new art of motion pictures to emerge as something quite different from a linear mix of theater plus photography. Most of what has been done up to now under the name of “educational technology” or “computers in education” is still at the stage of the linear mix of old instructional methods with new technologies. The topics I shall be discussing are some of the first probings toward a more organic interaction of fundamental educational principles and new methods for translating them into reality.
我们正处于教育史上可能出现彻底变革的时期，而这种变革的可能性与计算机的影响直接相关。今天，教育“市场”所提供的内容很大程度上取决于一个迟缓而保守的体系所能接受的内容。但这正是计算机的存在正在创造变革环境的过程中所处的位置。考虑一下在今天和不久的将来，一种新的教育理念可以在哪些条件下付诸实践。假设今天我有一个关于如何让孩子们更有效、更人性化地学习数学的想法。假设我已经能够说服一百万人相信这个想法是好的。对于许多产品来说，这样一个潜在的市场将保证成功。然而，在当今的教育界，这几乎没有什么影响力：全国一百万人仍然意味着每个城镇的学校系统中都是少数，因此可能没有有效的渠道让这百万人表达自己的声音。因此，不仅好的教育理念被束之高阁，而且发明过程本身也受到阻碍。这种对发明的抑制反过来又影响了从事教育的人选。很少有具有想象力、创造力和动力去创造伟大新发明的人进入这个领域。大多数进入的人很快就因沮丧而退出。教育界的保守主义已经成为一种自我延续的社会现象。
We are at a point in the history of education when radical change is possible, and the possibility for that change is directly tied to the impact of the computer. Today what is offered in the education “market” is largely determined by what is acceptable to a sluggish and conservative system. But this is where the computer presence is in the process of creating an environment for change. Consider the conditions under which a new educational idea can be put into practice today and in the near future. Let us suppose that today I have an idea about how children could learn mathematics more effectively and more humanely. And let us suppose that I have been able to persuade a million people that the idea is a good one. For many products such a potential market would guarantee success. Yet in the world of education today this would have little clout: A million people across the nation would still mean a minority in every town’s school system, so there might be no effective channel for the million voices to be expressed. Thus, not only do good educational ideas sit on the shelves, but the process of invention is itself stymied. This inhibition of invention in turn influences the selection of people who get involved in education. Very few with the imagination, creativity, and drive to make great new inventions enter the field. Most of those who do are soon driven out in frustration. Conservatism in the world of education has become a self-perpetuating social phenomenon.
幸运的是，恶性循环中有一个薄弱环节。在不久的将来，计算机将越来越多地成为个人的私有财产，这将逐渐将决定教育模式的权力交还给个人。教育将变得更加私人化，拥有好主意、不同主意、激动人心的想法的人将不再面临要么将自己的想法“卖”给保守的官僚机构，要么将其束之高阁的困境。他们将能够在开放的市场上直接向消费者提供这些想法。想象力和独创性将迎来新的机会。教育思想可能会复兴。
Fortunately, there is a weak link in the vicious circle. Increasingly, the computers of the very near future will be the private property of individuals, and this will gradually return to the individual the power to determine patterns of education. Education will become more of a private act, and people with good ideas, different ideas, exciting ideas will no longer be faced with a dilemma where they either have to “sell” their ideas to a conservative bureaucracy or shelve them. They will be able to offer them in an open marketplace directly to consumers. There will be new opportunities for imagination and originality. There might be a renaissance of thinking about education.
第二章
CHAPTER 2
数学恐惧症
Mathophobia
学习恐惧症
The Fear of Learning
柏拉图在他的门上写道：“只让几何学家进入。”时代变了。现在大多数想进入柏拉图思想世界的人既不懂数学，也感觉不到他们无视他的禁令有什么矛盾。我们文化在“人文”和“科学”之间的精神分裂支持了他们的安全感。柏拉图是一位哲学家，哲学属于人文学科，就像数学属于科学一样。
PLATO WROTE OVER HIS DOOR, “LET ONLY GEOMETERS ENTER.” TIMES have changed. Most of those who now seek to enter Plato’s intellectual world neither know mathematics nor sense the least contradiction in their disregard for his injunction. Our culture’s schizophrenic split between “humanities” and “science” supports their sense of security. Plato was a philosopher, and philosophy belongs to the humanities as surely as mathematics belongs to the sciences.
这种巨大的分歧已深深植根于我们的语言、世界观、社会组织、教育体系，甚至最近还植根于我们的神经生理学理论。这种分歧是自我延续的：文化越分化，双方在其新发展中就越会形成分裂。
This great divide is thoroughly built into our language, our worldview, our social organization, our educational system, and, most recently, even our theories of neurophysiology. It is self-perpetuating: The more the culture is divided, the more each side builds separation into its new growth.
我已经提出，计算机可能成为打破“两种文化”界限的力量。我知道人文学者可能会怀疑，一项“技术”能够改变他或她对哪种知识与他或她理解人的观点相关的假设。而对科学家来说，由于“优柔寡断”的人文主义思维的侵蚀而削弱了严谨性，其威胁性也丝毫不减。然而，我认为，计算机的存在可能会种下种子，长成一种不那么分离的文化认识论。
I have already suggested that the computer may serve as a force to break down the line between the “two cultures.” I know that the humanist may find it questionable that a “technology” could change his or her assumptions about what kind of knowledge is relevant to his or her perspective of understanding people. And to the scientist, dilution of rigor by the encroachment of “wishy-washy” humanistic thinking can be no less threatening. Yet the computer presence might, I think, plant seeds that could grow into a less dissociated cultural epistemology.
数学在当代文化中的地位是其分离的最明显症状之一。“人文”数学的出现，即人们认为与人类研究和“人文学科”没有区别的数学，很可能是即将发生改变的标志。因此，在这本书中，我试图展示计算机的存在如何使儿童与数学建立更人性化、更人道的关系。为了做到这一点，我必须超越数学的讨论。我必须对学习过程本身形成一种新的视角。
The status of mathematics in contemporary culture is one of the most acute symptoms of its dissociation. The emergence of a “humanistic” mathematics, one that is not perceived as separated from the study of man and “the humanities,” might well be the sign that a change is in sight. So in this book I try to show how the computer presence can bring children into a more humanistic as well as a more humane relationship with mathematics. In doing so I shall have to go beyond discussion of mathematics. I shall have to develop a new perspective on the process of learning itself.
聪明的成年人除了最基本的数学之外，在任何其他领域都成为自己无能的被动旁观者，这种情况并不罕见。个人可能会看到这种智力瘫痪的直接后果，即限制就业机会。但间接的次要后果甚至更加严重。大多数人在数学课上学到的主要教训之一是有严格的限制感。他们学到了人类知识的割据形象，他们认为这些知识是被不可逾越的铁幕隔开的一块块领土。我的挑战不是知识领域的主权，而是对它们之间轻松移动的限制。我不想把数学归结为文学，也不想把文学归结为数学。但我确实想说，它们各自的思维方式并不像通常认为的那样分离。因此，我使用数学国度的形象——数学将成为一种自然词汇——来发展我的想法，即计算机的存在可以将人文文化和数学/科学文化融合在一起。在这本书中，Mathland 是一个更大论证的第一步，该论证表明计算机的存在不仅可以改变我们教孩子数学的方式，而且从根本上改变我们的整个文化对知识和学习的看法。
It is not uncommon for intelligent adults to turn into passive observers of their own incompetence in anything but the most rudimentary mathematics. Individuals may see the direct consequences of this intellectual paralysis in terms of limiting job possibilities. But the indirect, secondary consequences are even more serious. One of the main lessons learned by most people in math class is a sense of having rigid limitations. They learn a balkanized image of human knowledge which they come to see as a patchwork of territories separated by impassable iron curtains. My challenge is not to the sovereignty of the intellectual territories but to the restrictions imposed on easy movement among them. I do not wish to reduce mathematics to literature or literature to mathematics. But I do want to argue that their respective ways of thinking are not as separate as is usually supposed. And so, I use the image of a Mathland—where mathematics would become a natural vocabulary—to develop my idea that the computer presence could bring the humanistic and mathematical/scientific cultures together. In this book, Mathland is the first step in a larger argument about how the computer presence can change not only the way we teach children mathematics but, much more fundamentally, the way in which our culture as a whole thinks about knowledge and learning.
在我看来，“数学恐惧症”一词有两种含义。一种是普遍的对数学的恐惧，这种恐惧往往与真正的恐惧症一样强烈。另一种含义来自“数学”一词的词干含义。在希腊语中，它表示一般意义上的“学习”。在我们的文化中，对学习的恐惧与对数学的恐惧一样普遍（尽管更经常被伪装）。孩子们在一生中都是热切而能干的学习者。他们必须学会在学习一般知识和数学知识方面遇到困难。在“数学”的这两种含义中，都存在着从数学爱好者到数学恐惧者的转变，从热爱数学和学习的人到对两者都感到恐惧的人。我们将研究这种转变是如何发生的，并提出一些想法，即计算机的存在如何能够抵消这种转变。首先，让我来思考一下儿童学习的感受。
To my ear the word “mathophobia” has two associations. One of these is a widespread fear of mathematics, which often has the intensity of a real phobia. The other comes from the meaning of the stem “math.” In Greek it means “learning” in a general sense.I In our culture, fear of learning is no less endemic (although more frequently disguised) than fear of mathematics. Children begin their lives as eager and competent learners. They have to learn to have trouble with learning in general and mathematics in particular. In both senses of “math” there is a shift from mathophile to mathophobe, from lover of mathematics and of learning to a person fearful of both. We shall look at how this shift occurs and develop some idea of how the computer presence could serve to counteract it. Let me begin with some reflections on what it is like to learn as a child.
儿童学习能力强，这一点对大多数人来说都是显而易见的，以至于他们认为几乎不值得记录。其中一个很明显的学习率高的领域是口语词汇的习得。两岁时，很少有孩子能掌握超过几百个单词。到四年后上一年级时，他们已经掌握了数千个单词。显然，他们每天都在学习许多新单词。
That children learn a great deal seems so obvious to most people that they believe it is scarcely worth documenting. One area in which a high rate of learning is very plain is the acquisition of a spoken vocabulary. At age two very few children have more than a few hundred words. By the time they enter first grade, four years later, they know thousands of words. They are evidently learning many new words every day.
虽然我们可以“看到”孩子们学习单词，但很难发现他们学习数学的速度与孩子相似甚至更快。但这正是皮亚杰毕生研究儿童知识起源的结果。他的发现的一个更微妙的后果是，成年人未能认识到儿童学习的范围和性质，因为我们认为理所当然的知识结构使许多学习变得不可见。我们在后来被称为皮亚杰“守恒定律”的现象中可以最清楚地看到这一点（见图2）。
While we can “see” that children learn words, it is not quite as easy to see that they are learning mathematics at a similar or greater rate. But this is precisely what has been shown by Piaget’s lifelong study of the genesis of knowledge in children. One of the more subtle consequences of his discoveries is the revelation that adults fail to appreciate the extent and the nature of what children are learning, because knowledge structures we take for granted have rendered much of that learning invisible. We see this most clearly in what have come to be known as Piagetian “conservations” (see Figure 2).
图 2. 液体守恒
Figure 2. The Conservation of Liquids
对于成年人来说，将液体从一个杯子倒入另一个杯子显然不会改变体积（忽略溢出或残留的液滴等微小影响）。体积守恒是如此明显，以至于在皮亚杰之前似乎没有人想到四岁的孩子可能根本就觉得不明显。II孩子们需要相当大的智力发展才能形成“自然保护主义”的世界观。体积守恒只是他们都会学习的众多守恒之一。另一个是数字守恒。同样，大多数成年人没有想到孩子必须学会以不同的顺序计数一组物体应该得到相同的结果。对于成年人来说，计数只是一种确定“有多少个物体”的方法。操作的结果是与计数行为无关的“客观事实”。但是，将数字与计数（将产品与过程）分开是基于认识论的预设，而这些预设不仅不为前自然保护主义儿童所知，而且也与他们的世界观格格不入。这些守恒定律只是孩子们自学的庞大“隐藏”数学知识结构的一部分。在四五岁孩子的直观几何学中，直线不一定是两点之间的最短距离，两点之间慢走也不一定比快走花费更多时间。在这里，同样，缺失的不仅仅是知识的“项目”，还有将“最短”作为路径属性而不是穿越路径行为属性这一概念所依据的认识论前提。
For an adult it is obvious that pouring liquid from one glass to another does not change the volume (ignoring such little effects as drops that spilled or remained behind). The conservation of volume is so obvious that it seems not to have occurred to anyone before Piaget that children of four might not find it obvious at all.II A substantial intellectual growth is needed before children develop the “conservationist” view of the world. The conservation of volume is only one of many conservations they all learn. Another is the conservation of numbers. Again, it does not occur to most adults that a child must learn that counting a collection of objects in a different order should yield the same result. For adults counting is simply a method of determining how many objects “there are.” The result of the operation is an “objective fact” independent of the act of counting. But the separation of number from counting (of product from process) rests on epistemological presuppositions not only unknown to preconservationist children but alien to their worldview. These conservations are only part of a vast structure of “hidden” mathematical knowledge that children learn by themselves. In the intuitive geometry of the child of four or five, a straight line is not necessarily the shortest distance between two points, and walking slowly between two points does not necessarily take more time than walking fast. Here, too, it is not merely the “item” of knowledge that is missing, but the epistemological presupposition underlying the idea of “shortest” as a property of the path rather than of the action of traversing it.
所有这些不应被理解为儿童缺乏知识。皮亚杰已经证明了幼儿如何持有他们自己认为完全一致的世界理论。这些理论是所有儿童自发“学习”的，具有成熟的组成部分，虽然表达了不同的数学，但与我们（成人）文化中普遍接受的数学相比，它们并不缺乏“数学性”。隐藏的学习过程至少有两个阶段：早在学龄前，每个孩子就首先构建了一个或多个成人前的世界理论，然后转向更像成人的观点。所有这些都是通过我所说的皮亚杰学习来完成的，这种学习过程具有许多学校应该羡慕的特点：它很有效（所有孩子都能达到这种水平），它很便宜（它似乎不需要老师或课程开发），它很人性化（孩子们似乎以无忧无虑的精神去做这件事，没有明确的外部奖励和惩罚）。
None of this should be understood as mere lack of knowledge on the part of the children. Piaget has demonstrated how young children hold theories of the world that, in their own terms, are perfectly coherent. These theories, spontaneously “learned” by all children, have well-developed components that are not less “mathematical,” though expressing a different mathematics, than the one generally accepted in our (adult) culture. The hidden learning process has at least two phases: Already in the preschool years every child first constructs one or more preadult theorizations of the world and then moves toward more adultlike views. And all this is done through what I have called Piagetian learning, a learning process that has many features the schools should envy: It is effective (all the children get there), it is inexpensive (it seems to require neither teacher nor curriculum development), and it is humane (the children seem to do it in a carefree spirit without explicit external rewards and punishments).
我们社会中的成年人在多大程度上失去了儿童对学习的积极态度因人而异。人口中有一个未知但肯定相当大比例的人几乎完全放弃了学习。这些人很少，甚至从未进行过刻意学习，他们认为自己既不擅长学习，也不可能喜欢学习。社会和个人的代价是巨大的：数学恐惧症会在文化和物质上限制人们的生活。更多的人没有完全放弃学习，但仍然严重受到对自己能力根深蒂固的消极信念的阻碍。缺陷成为身份：“我学不会法语，我对语言没有天赋”；“我永远不可能成为一名商人，我不擅长数字”；“我不会平行滑雪，我从来没有协调过。”这些信念经常像迷信一样被仪式性地重复。而且，就像迷信一样，它们创造了一个禁忌的世界：在这种情况下，就是学习禁忌。在本章和第 3 章中，我们讨论了一些实验，这些实验表明，这些自我形象往往与非常有限的现实相对应——通常是与一个人的“学校现实”相对应。在具有适当的情感和智力支持的学习环境中，“不协调”的人可以学习杂耍等马戏艺术，而那些“不懂数字”的人不仅可以学习数学，还可以享受数学的乐趣。
The extent to which adults in our society have lost the child’s positive stance toward learning varies from individual to individual. An unknown but certainly significant proportion of the population has almost completely given up on learning. These people seldom, if ever, engage in deliberate learning and see themselves as neither competent at it nor likely to enjoy it. The social and personal cost is enormous: Mathophobia can, culturally and materially, limit people’s lives. Many more people have not completely given up on learning but are still severely hampered by entrenched negative beliefs about their capacities. Deficiency becomes identity: “I can’t learn French, I don’t have an ear for languages”; “I could never be a businessman, I don’t have a head for figures”; “I can’t get the hang of parallel skiing, I never was coordinated.” These beliefs are often repeated ritualistically, like superstitions. And, like superstitions, they create a world of taboos: in this case, taboos on learning. In this chapter and chapter 3, we discuss experiments that demonstrate that these self-images often correspond to a very limited reality—usually to a person’s “school reality.” In a learning environment with the proper emotional and intellectual support, the “uncoordinated” can learn circus arts like juggling and those with “no head for figures” learn not only that they can do mathematics but that they can enjoy it as well.
虽然这些负面的自我形象是可以克服的，但在个人生活中，它们却极其顽固，并且具有强大的自我强化能力。如果人们坚信自己无法做数学，他们通常就能成功阻止自己做任何他们认为是数学的事情。这种自我破坏的后果是个人失败，而每次失败都会强化最初的信念。当这种信念不仅由个人持有，而且由我们的整个文化持有时，其危害可能最大。
Although these negative self-images can be overcome, in the life of an individual they are extremely robust and powerfully self-reinforcing. If people believe firmly enough that they cannot do math, they will usually succeed in preventing themselves from doing whatever they recognize as math. The consequences of such self-sabotage is personal failure, and each failure reinforces the original belief. And such beliefs may be most insidious when held not only by individuals but by our entire culture.
我们的孩子成长于一种文化中，这种文化中充斥着“聪明人”和“愚笨人”的观念。社会将个人视为一系列才能。有些人“擅长数学”，有些人“不会数学”。一切都为孩子们设定好了，让他们将第一次不成功或不愉快的学习经历归咎于自己的缺陷。因此，孩子们认为失败要么将他们归入“愚笨人”群体，要么更常见的是归入“在 x 方面愚笨”的群体（正如我们所指出的，x 通常等于数学）。在这个框架下，孩子们会根据自己的局限性来定义自己，这种定义将在他们的一生中得到巩固和强化。只有极少数的特殊事件才会促使人们重新组织他们的智力自我形象，从而开辟新的视角来了解什么是可以学习的。
Our children grow up in a culture permeated with the idea that there are “smart people” and “dumb people.” The social construction of the individual is as a bundle of aptitudes. There are people who are “good at math” and people who “can’t do math.” Everything is set up for children to attribute their first unsuccessful or unpleasant learning experiences to their own disabilities. As a result, children perceive failure as relegating them either to the group of “dumb people” or, more often, to a group of people “dumb at x” (where, as we have pointed out, x often equals mathematics). Within this framework children will define themselves in terms of their limitations, and this definition will be consolidated and reinforced throughout their lives. Only rarely does some exceptional event lead people to reorganize their intellectual self-image in such a way as to open up new perspectives on what is learnable.
这种关于人类能力结构的信念并不容易被破坏。根除流行的信念从来都不是件容易的事。但这里的困难还因其他几个因素而加剧。首先，关于人类能力的流行理论似乎得到了“科学”理论的支持。毕竟，心理学家谈论的是测量能力。但我们想象数学世界的简单思想实验严重质疑了测量的意义。
This belief about the structure of human abilities is not easy to undermine. It is never easy to uproot popular beliefs. But here the difficulty is compounded by several other factors. First, popular theories about human aptitudes seem to be supported by “scientific” ones. After all, psychologists talk in terms of measuring aptitudes. But the significance of what is measured is seriously questioned by our simple thought experiment of imagining Mathland.
尽管想象一个数学乐园的思想实验没有回答如何真正创建一个数学乐园的问题，但它完全可以严谨地证明，关于数学才能的公认信念不是从现有证据中得出的。1但是，由于真正恐数学的读者可能难以自己进行这个实验，所以我将换一种形式来强化这个论点。想象一下，孩子们被迫每天花一个小时在方格纸上画舞步，并且必须通过这些“舞蹈事实”测试才能开始跳舞。我们难道不会认为这个世界充满了“恐舞者”吗？我们会说那些走进舞池、听着音乐的人最有“舞蹈天赋”吗？在我看来，仅仅因为孩子们不愿意花数百小时做算术就得出关于数学才能的结论，并不合适。
Although the thought experiment of imagining a Mathland leaves open the question of how a Mathland can actually be created, it is completely rigorous as a demonstration that the accepted beliefs about mathematical aptitude do not follow from the available evidence.1 But since truly mathophobic readers might have trouble making this experiment their own, I shall reinforce the argument by casting it in another form. Imagine that children were forced to spend an hour a day drawing dance steps on squared paper and had to pass tests in these “dance facts” before they were allowed to dance physically. Would we not expect the world to be full of “dancophobes”? Would we say that those who made it to the dance floor and music had the greatest “aptitude for dance”? In my view, it is no more appropriate to draw conclusions about mathematical aptitude from children’s unwillingness to spend many hundreds of hours doing sums.
人们可能希望，如果我们从寓言转向更严格的心理学方法，我们就能获得一些关于个人真正能力上限问题的“更难”数据。但事实并非如此：当代教育心理学使用的范式侧重于研究孩子们在我们生活的“反数学世界”中如何学习或（更常见的是）不学习数学。这种研究的方向与以下寓言类似：
One might hope that if we pass from parables to the more rigorous methods of psychology we could get some “harder” data on the problem of the true ceilings of competence attainable by individuals. But this is not so: The paradigm in use by contemporary educational psychology is focused on investigations of how children learn or (more usually) don’t learn mathematics in the “anti-Mathland” in which we all live. The direction of such research has an analogy in the following parable:
想象一下，有人生活在十九世纪，他觉得有必要改进交通方式。他坚信，通往新方法的途径始于对现有问题的深刻理解。于是，他开始仔细研究马车之间的差异。他用最精妙的方法仔细记录了速度如何随着各种车轴、轴承和驾驭技术的形式和材质而变化。
Imagine someone living in the nineteenth century who felt the need to improve methods of transportation. He was persuaded that the route to new methods started with a deep understanding of the existing problems. So he began a careful study of the differences among horse-drawn carriages. He carefully documented by the most refined methods how speed varied with the form and substance of various kinds of axles, bearings, and harnessing techniques.
回想起来，我们知道十九世纪交通工具的发展道路截然不同。汽车和飞机的发明并非源于对马车等前身如何工作或不工作进行详细研究。然而，这是当代教育研究的模式。教育研究的标准范式以现有的课堂或课外文化为主要研究对象。许多研究都涉及数学或科学学生从当今的学校教育中获得的糟糕观念。甚至有一种非常流行的“人文主义”观点认为，“好的”教学法应该以这些糟糕的思维方式为出发点。人们很容易同情这种人文主义的意图。然而，我认为这一策略意味着致力于维护传统体系。这类似于改进马车的车轴。但真正的问题是，我们能否发明“教育汽车”。由于教育心理学尚未探讨这个问题（本书的中心主题），我们必须得出结论：关于能力倾向的信念的“科学”基础确实非常不可靠。但这些信念在学校、考试制度和大学录取标准中制度化，因此，它们的社会基础牢固，而科学基础薄弱。
In retrospect, we know that the road that led from nineteenth-century transportation was quite different. The invention of the automobile and the airplane did not come from a detailed study of how their predecessors, such as horse-drawn carriages, worked or did not work. Yet, this is the model for contemporary educational research. The standard paradigms for education research take the existing classroom or extracurricular culture as the primary object of study. There are many studies concerning the poor notions of math or science students acquire from today’s schooling. There is even a very prevalent “humanistic” argument that “good” pedagogy should take these poor ways of thinking as its starting point. It is easy to sympathize with the humane intent. Nevertheless I think that the strategy implies a commitment to preserving the traditional system. It is analogous to improving the axle of the horse-drawn cart. But the real question, one might say, is whether we can invent the “educational automobile.” Since this question (the central theme of this book) has not been addressed by educational psychology, we must conclude that the “scientific” basis for beliefs about aptitudes is really very shaky. But these beliefs are institutionalized in schools, in testing systems, and in college admissions criteria, and consequently, their social basis is as firm as their scientific basis is weak.
从幼儿园开始，孩子们就要接受语言和数学能力测试，这两项能力被认为是“真实的”和可分离的实体。这些测试的结果将作为一揽子能力进入每个孩子社会建构。一旦约翰尼和他的老师对约翰尼有了共同的看法，认为他“擅长”艺术而“不擅长”数学，这种看法就会根深蒂固。这一点在当代教育心理学中被广泛接受。但学校建构能力的方式还有更深层次的方面。以我观察过的八九岁孩子为例。吉姆是一个来自职业家庭的孩子，他非常擅长语言，但对数学有恐惧感。他对语言和说话的热爱很早就表现出来了，早在他上学之前。数学恐惧症是在学校里形成的。我认为，这是他语言早熟的直接结果。我从他的父母那里得知，吉姆很早就养成了用语言描述自己正在做的事情的习惯，通常是大声描述。这种习惯让他与父母和幼儿园老师之间产生了一些小矛盾。真正的麻烦出现在他上算术课的时候。到那时，他已经学会了控制“大声说话”，但我相信他仍然保持着内心对自己活动的持续评论。在数学课上，他感到很为难：他根本不知道如何谈论做算术。他缺乏词汇量（就像我们大多数人一样）和目标感。由于对他的言语习惯感到沮丧，他开始讨厌数学，而这种仇恨又导致了后来测试证实的数学能力差。
From kindergarten on, children are tested for verbal and quantitative aptitudes, conceived of as “real” and separable entities. The results of these tests enter into the social construction of each child as a bundle of aptitudes. Once Johnny and his teacher have a shared perception of Johnny as a person who is “good at” art and “poor at” math, this perception has a strong tendency to dig itself in. This much is widely accepted in contemporary educational psychology. But there are deeper aspects to how school constructs aptitudes. Consider the case of a child I observed through his eighth and ninth years. Jim was a highly verbal and mathophobic child from a professional family. His love for words and for talking showed itself very early, long before he went to school. The mathophobia developed at school. My theory is that it came as a direct result of his verbal precocity. I learned from his parents that Jim had developed an early habit of describing in words, often aloud, whatever he was doing as he did it. This habit caused him minor difficulties with parents and preschool teachers. The real trouble came when he hit the arithmetic class. By this time he had learned to keep “talking aloud” under control, but I believe that he still maintained his inner running commentary on his activities. In his math class he was stymied: He simply did not know how to talk about doing sums. He lacked a vocabulary (as most of us do) and a sense of purpose. Out of this frustration of his verbal habits grew a hatred of math, and out of the hatred grew what the tests later confirmed as poor aptitude.
对我来说，这个故事令人心酸。我相信，智力上的弱点往往源自智力上的优势，就像吉姆的情况一样。而且，并非只有语言上的优势会削弱其他方面的优势。每一个细心观察儿童的人都一定看到过类似的过程在不同方向上发挥作用：例如，一个迷恋逻辑顺序的孩子会对英语拼写感到厌烦，并从此发展出对写作的厌恶。
For me the story is poignant. I am convinced that what shows up as intellectual weakness very often grows, as Jim’s did, out of intellectual strengths. And it is not only verbal strengths that undermine others. Every careful observer of children must have seen similar processes working in different directions: For example, a child who has become enamored of logical order is set up to be turned off by English spelling and to go on from there to develop a global dislike for writing.
数学乐园的概念展示了如何使用计算机作为工具来摆脱吉姆和他患有诵读困难症的同伴的处境。这两个孩子都是我们文化中语言和数学之间严格分离的受害者。在本章中我们将要描述的数学乐园中，吉姆对语言的热爱和技能可以被用来服务于他正式的数学发展，而不是与之对立，而另一个孩子对逻辑的热爱可以被用来服务于对语言学兴趣的发展。
The Mathland concept shows how to use computers as vehicles to escape from the situation of Jim and his dyslexic counterpart. Both children are victims of our culture’s hard-edged separation between the verbal and the mathematical. In the Mathland we shall describe in this chapter, Jim’s love and skill for language could be mobilized to serve his formal mathematical development instead of opposing it, and the other child’s love for logic could be recruited to serve the development of interest in linguistics.
调动儿童的多种优势，使其服务于所有智力活动领域，这一概念是对以下观点的回答：不同的能力可能反映出大脑发育的实际差异。人们普遍认为，大脑或大脑中存在不同的“器官”，分别负责数学和语言。根据这种思维方式，儿童根据哪个大脑器官最强，分为语言能力强的儿童和数学能力强的儿童。但从解剖学到智力的论点反映了一系列认识论假设。例如，它假设只有一条通往数学的路线，如果这条路线“在解剖学上被阻塞”，孩子就无法到达目的地。事实上，对于当代社会中的大多数儿童来说，通往“高级”数学的路线可能确实只有一条，即通过学校数学。但即使进一步的脑生物学研究证实这条路线依赖于一些儿童可能缺失的解剖学大脑器官，也不能得出数学本身依赖于这些大脑器官的结论。相反，我们应该寻找其他途径。由于这本书论证了替代途径确实存在，因此可以将其解读为表明功能对大脑的依赖本身就是一种社会建构。
The concept of mobilizing a child’s multiple strengths to serve all domains of intellectual activity is an answer to the suggestion that differing aptitudes may reflect actual differences in brain development. It has become commonplace to talk as if there are separate brains, or separate “organs” in the brain, for mathematics and for language. According to this way of thinking, children split into the verbally and the mathematically apt depending on which brain organs are strongest. But the argument from anatomy to intellect reflects a set of epistemological assumptions. It assumes, for example, that there is only one route to mathematics and that if this route is “anatomically blocked,” the child cannot get to the destination. Now, in fact, for most children in contemporary societies there may indeed be only one route into “advanced” mathematics, the route via school math. But even if further research in brain biology confirms that this route depends on anatomical brain organs that might be missing in some children, it would not follow that mathematics itself is dependent on these brain organs. Rather, it would follow that we should seek out other routes. Since this book is an argument that alternate routes do exist, it can be read as showing how the dependency of function on the brain is itself a social construct.
为了论证的目的，我们假设大脑中有一个特殊区域特别擅长进行我们在学校教给孩子的数字心理操作，我们把它称为MAD，即“数学习得装置”。2基于这一假设，人类在历史进程中进化出充分利用 MAD 的算术教学方法，这是有道理的。但是，虽然这些方法对大多数人有效，对整个社会也有效，但对于MAD因其他原因（可能是“神经质”）受损或无法使用的人来说，依赖这些方法将是灾难性的。这样的人会在学校失败并被诊断为“计算障碍”的受害者。只要我们坚持让孩子们通过标准途径学习算术，我们就会继续通过客观测试“证明”这些孩子真的不能“做算术”。但这就像证明聋哑儿童因为听不见而不能说话一样。正如手语使用手和眼睛来绕过更常见的说话器官一样，绕过MAD的其他数学运算方法即使与通常的方法不同，也可能与通常的方法一样好。
Let us grant, for the sake of argument, that there is a special part of the brain especially good at performing the mental manipulations of numbers we teach children in school, and let’s call it the MAD, or “math acquisition device.”2 On this assumption it would make sense that in the course of history humankind would have evolved methods of doing and of teaching arithmetic that take full advantage of the MAD. But while these methods would work for most of us, and so for society as a whole, reliance on them would be catastrophic for an individual whose MAD happened to be damaged or inaccessible for some other (perhaps “neurotic”) reason. Such a person would fail at school and be diagnosed as a victim of “dyscalculia.” And as long as we insist on making children learn arithmetic by the standard route, we will continue to “prove” by objective tests that these children really cannot “do arithmetic.” But this is like proving that deaf children cannot have language because they don’t hear. Just as sign languages use hands and eyes to bypass the more usual speaking organs so, too, alternative ways of doing mathematics that bypass the MAD may be as good as, even if different from, the usual ones.
但我们不必求助于神经学来解释为什么有些孩子无法熟练掌握数学。没有音乐或舞池的舞蹈课的类比是严肃的。我们的教育文化为数学学习者提供了稀缺的资源来理解他们所学的内容。结果，我们的孩子被迫遵循最糟糕的数学学习模式。这是死记硬背的模式，材料被视为毫无意义；这是一个分离的模式。我们在教授更具文化融合性的数学方面遇到的一些困难是由于一个客观问题：在我们拥有计算机之前，最基本、最吸引人的数学与日常生活中根深蒂固的东西之间几乎没有很好的联系点。但计算机——一个在家庭、学校和工作场所的日常生活中讲数学的东西——能够提供这样的联系。教育面临的挑战是找到利用它们的方法。
But we do not have to appeal to neurology to explain why some children do not become fluent in mathematics. The analogy of the dance class without music or dance floor is a serious one. Our education culture gives mathematics learners scarce resources for making sense of what they are learning. As a result our children are forced to follow the very worst model for learning mathematics. This is the model of rote learning, where material is treated as meaningless; it is a dissociated model. Some of our difficulties in teaching a more culturally integrable mathematics have been due to an objective problem: Before we had computers there were very few good points of contact between what is most fundamental and engaging in mathematics and anything firmly planted in everyday life. But the computer—a mathematics-speaking being in the midst of the everyday life of the home, school, and workplace—is able to provide such links. The challenge to education is to find ways to exploit them.
数学当然不是分离学习的唯一例子。但它是一个很好的例子，原因恰恰在于许多读者现在可能希望我谈论其他东西。我们的文化如此仇视数学，如此害怕数学，如果我能证明计算机如何将我们带入与数学的新关系，我就有充分的理由声称计算机有能力改变我们与可能害怕的其他学习类型之间的关系。数学领域的体验，例如进入“数学对话”，让个人感受到做各种以前可能看起来“太难”的事情的可能性。从这个意义上说，与计算机的接触可以为人们打开获取知识的渠道，不是通过向他们提供经过处理的信息，而是通过挑战他们对自己做出的一些限制性假设。
Mathematics is certainly not the only example of dissociated learning. But it is a very good example for precisely the reason that many readers are probably now wishing that I would talk about something else. Our culture is so mathophobic, so math-fearing, that if I could demonstrate how the computer can bring us into a new relationship to mathematics, I would have a strong foundation for claiming that the computer has the ability to change our relation to other kinds of learning we might fear. Experiences in Mathland, such as entering into a “mathematical conversation,” give the individual a liberating sense of the possibilities of doing a variety of things that may have previously seemed “too hard.” In this sense, contact with the computer can open access to knowledge for people, not instrumentally by providing them with processed information, but by challenging some constraining assumptions they make about themselves.
我提议的基于计算机的数学世界将自然的皮亚杰学习（皮亚杰学习，用于解释儿童学习第一语言的过程）扩展到学习数学。皮亚杰学习通常深深植根于其他活动中。例如，婴儿没有专门用于“学习说话”的时间。这种学习模式与分离式学习相反，分离式学习是相对与其他类型的活动（心理和身体活动）分离进行的学习。在我们的文化中，学校的数学教学是分离式学习的典型。对于大多数人来说，数学是作为药物来教授和服用的。在数学分离方面，我们的文化最接近于讽刺其自身最糟糕的认识论异化习惯。在LOGO环境中，我们已经模糊了一些界限：没有将任何特定的计算机活动专门列为“学习数学”。
The computer-based Mathland I propose extends the kind of natural, Piagetian learning that accounts for children’s learning a first language to learning mathematics. Piagetian learning is typically deeply embedded in other activities. For example, the infant does not have periods set aside for “learning talking.” This model of learning stands in opposition to dissociated learning, learning that takes place in relative separation from other kinds of activities, mental and physical. In our culture, the teaching of mathematics in schools is paradigmatic of dissociated learning. For most people, mathematics is taught and taken as medicine. In its dissociation of mathematics, our culture comes closest to caricaturing its own worst habits of epistemological alienation. In LOGO environments we have done some blurring of boundaries: No particular computer activities are set aside as “learning mathematics.”
让学习者“理解”数学的问题涉及到让“形式描述”语言变得有意义的更普遍的问题。因此，在讨论计算机如何帮助赋予数学意义之前，我们将看几个例子，在这些例子中，计算机帮助赋予人们通常不认为是数学的知识领域中的形式描述语言意义。在我们的第一个例子中，这个领域是语法，对许多人来说，语法的威胁性只比数学小一点。
The problem of making mathematics “make sense” to the learner touches on the more general problem of making a language of “formal description” make sense. So before turning to examples of how the computer helps give meaning to mathematics, we shall look at several examples where the computer helped give meaning to a language of formal description in domains of knowledge that people do not usually count as mathematics. In our first example, the domain is grammar, for many people a subject only a little less threatening than math.
在一项为期一年的研究中，一群“普通”七年级学生将功能强大的计算机带入课堂，学生们正在创作所谓的“计算机诗歌”。他们使用计算机程序来生成句子。他们为计算机提供了一个句法结构，让计算机从给定的单词列表中随机选择。结果就是我们在下图中看到的那种具体的诗歌。其中一名名叫珍妮的十三岁学生在第一天开始使用计算机工作时问道：“为什么我们被选中？我们不是大脑。”这深深地打动了项目工作人员。这项研究有意选择了学习成绩“普通”的孩子。有一天，珍妮非常兴奋地走进来。她发现了一个问题。“现在我知道为什么我们有名词和动词了，”她说。在学校里，珍妮多年来一直在学习语法类别。她从来没有理解过名词、动词和副词之间的区别。但现在很明显，她在语法方面的困难并不是由于无法运用逻辑范畴。而是别的原因。她只是觉得这门课毫无意义。她无法理解语法的意义，无法理解语法的用途。当她问起语法的用途时，老师们的解释似乎明显不诚实。她说老师告诉她“语法能帮助你更好地说话。”
Well into a yearlong study that put powerful computers in the classrooms of a group of “average” seventh graders, the students were at work on what they called “computer poetry.” They were using computer programs to generate sentences. They gave the computer a syntactic structure within which to make random choices from given lists of words. The result is the kind of concrete poetry we see in the illustration that follows. One of the students, a thirteen-year-old named Jenny, had deeply touched the project’s staff by asking on the first day of her computer work, “Why were we chosen for this? We’re not the brains.” The study had deliberately chosen children of “average” school performance. One day Jenny came in very excited. She had made a discovery. “Now I know why we have nouns and verbs,” she said. For many years in school Jenny had been drilled in grammatical categories. She had never understood the differences between nouns and verbs and adverbs. But now it was apparent that her difficulty with grammar was not due to an inability to work with logical categories. It was something else. She had simply seen no purpose in the enterprise. She had not been able to make any sense of what grammar was about in the sense of what it might be for. And when she had asked what it was for, the explanations that her teachers gave seemed manifestly dishonest. She said she had been told that “grammar helps you talk better.”
疯狂的智障让甜蜜的史努比尖叫
INSANE RETARD MAKES BECAUSE SWEET SNOOPY SCREAMS
性感狼喜欢这就是性感女士讨厌的原因
SEXY WOLF LOVES THATS WHY THE SEXY LADY HATES
丑狗恨，丑人爱
UGLY MAN LOVES BECAUSE UGLY DOG HATES
疯狼讨厌因为疯狼跳过
MAD WOLF HATES BECAUSE INSANE WOLF SKIPS
性感白痴尖叫 这就是为什么性感白痴
SEXY RETARD SCREAMS THATS WHY THE SEXY RETARD
讨厌
HATES
瘦史努比因为胖狼跳而奔跑
THIN SNOOPY RUNS BECAUSE FAT WOLF HOPS
甜美的福吉尼跳过一个胖女人跑步
SWEET FOGINY SKIPS A FAT LADY RUNS
珍妮的具体诗
Jenny’s Concrete Poetry
事实上，要追溯学习语法和提高口语能力之间的联系，就需要对复杂的语言学习过程有更远的眼光，而珍妮在第一次接触语法的时候不可能有这样的眼光。她当然没有看到语法对说话有什么帮助，也不认为她的说话需要任何帮助。因此，她学会了带着怨恨去学习语法。而且，就像我们大多数人的情况一样，怨恨注定会失败。但现在，当她试图让电脑创作诗歌时，发生了一件了不起的事情。她发现自己把单词分类了，不是因为她被告知必须这样做，而是因为她需要这样做。为了“教”她的电脑制作看起来像英语的字符串，她必须“教”它选择适当类别的单词。她从这次与机器的体验中学到的语法绝不是机械的或例行公事。她的学习是深刻而有意义的。珍妮所做的不仅仅是学习特定语法类别的定义。她理解了这样一个基本概念：单词（就像事物一样）可以放在不同的组或集合中，而且这样做对她有用。她不仅“理解”了语法，还改变了她与语法的关系。语法是“她的”，在她使用电脑的那一年里，像这样的事件帮助珍妮改变了她对自己的印象。她的表现也发生了变化；她之前成绩很低，在她剩下的几年里成绩都是“全优”。她明白了她毕竟可以成为一个“有头脑的人”。
In fact, tracing the connection between learning grammar and improving speech requires a more distanced view of the complex process of learning language than Jenny could have been given at the age she first encountered grammar. She certainly didn’t see any way in which grammar could help talking, nor did she think her talking needed any help. Therefore she learned to approach grammar with resentment. And, as is the case for most of us, resentment guaranteed failure. But now, as she tried to get the computer to generate poetry, something remarkable happened. She found herself classifying words into categories, not because she had been told she had to but because she needed to. In order to “teach” her computer to make strings of words that would look like English, she had to “teach” it to choose words of an appropriate class. What she learned about grammar from this experience with a machine was anything but mechanical or routine. Her learning was deep and meaningful. Jenny did more than learn definitions for particular grammatical classes. She understood the general idea that words (like things) can be placed in different groups or sets, and that doing so could work for her. She not only “understood” grammar, she changed her relationship to it. It was “hers,” and during her year with the computer, incidents like this helped Jenny change her image of herself. Her performance changed too; her previously low to average grades became “straight A’s” for her remaining years of school. She learned that she could be “a brain” after all.
很容易理解，为什么数学和语法对孩子来说毫无意义，因为周围的人都无法理解它们，为什么帮助孩子理解它们需要的不仅仅是老师发表正确的演讲或在黑板上画出正确的图表。我问过许多老师和家长，他们认为数学是什么，为什么学习数学很重要。很少有人对数学有足够连贯的看法，足以证明孩子一生中花几千个小时来学习数学是合理的，孩子们也感觉到了这一点。当老师告诉学生花这么多时间学习算术是为了能够在超市检查零钱时，老师根本不会相信。孩子们把这种“理由”看作是成人含糊其辞的又一个例子。当孩子们被告知学校数学“有趣”时，也会产生同样的效果，因为他们很确定说数学的老师会把闲暇时间花在除了这种所谓充满乐趣的活动之外的任何事情上。告诉他们需要数学才能成为科学家也无济于事——大多数孩子没有这样的计划。孩子们可以清楚地看到，老师并不比他们更喜欢数学，而这样做的原因仅仅是因为数学已经被写进了课程。所有这些都削弱了孩子们对成人世界和教育过程的信心。我认为这给教育关系带来了深深的不诚实因素。
It is easy to understand why math and grammar fail to make sense to children when they fail to make sense to everyone around them and why helping children to make sense of them requires more than a teacher making the right speech or putting the right diagram on the board. I have asked many teachers and parents what they thought mathematics to be and why it was important to learn it. Few held a view of mathematics that was sufficiently coherent to justify devoting several thousand hours of a child’s life to learning it, and children sense this. When a teacher tells a student that the reason for those many hours of arithmetic is to be able to check the change at the supermarket, the teacher is simply not believed. Children see such “reasons” as one more example of adult double talk. The same effect is produced when children are told school math is “fun” when they are pretty sure that teachers who say so spend their leisure hours on anything except this allegedly fun-filled activity. Nor does it help to tell them that they need math to become scientists—most children don’t have such a plan. The children can see perfectly well that the teacher does not like math any more than they do and that the reason for doing it is simply that it has been inscribed into the curriculum. All of this erodes children’s confidence in the adult world and the process of education. And I think it introduces a deep element of dishonesty into the educational relationship.
孩子们认为学校关于数学的说辞是模棱两可的。为了纠正这种情况，我们必须首先承认孩子的看法从根本上是正确的。学校强加给孩子们的数学既无意义，又无趣，甚至没有多大用处。这并不意味着个别孩子不能把它变成一种有价值和令人愉快的个人游戏。对一些人来说，游戏就是得分；对另一些人来说，游戏就是智胜老师和系统。对许多人来说，学校数学的重复性是令人愉快的，正是因为它如此不费脑力和脱节，为人们提供了一个避难所，使他们不必去思考课堂上发生的事情。但这一切都证明了孩子们的聪明才智。说学校数学虽然本质上很枯燥，但有创造力的孩子可以从中找到兴奋和意义，这并不是为学校数学辩解。
Children perceive the school’s rhetoric about mathematics as double talk. In order to remedy the situation we must first acknowledge that the child’s perception is fundamentally correct. The kind of mathematics foisted on children in schools is not meaningful, fun, or even very useful. This does not mean that an individual child cannot turn it into a valuable and enjoyable personal game. For some the game is scoring grades; for others it is outwitting the teacher and the system. For many, school math is enjoyable in its repetitiveness, precisely because it is so mindless and dissociated that it provides a shelter from having to think about what is going on in the classroom. But all this proves is the ingenuity of children. It is not a justification for school math to say that despite its intrinsic dullness, inventive children can find excitement and meaning in it.
重要的是要记住数学与数学或学校数学之间的区别。数学是一个广阔的研究领域，它的美妙之处大多数非数学家很少察觉到。
It is important to remember the distinction between mathematics—a vast domain of inquiry whose beauty is rarely suspected by most nonmathematicians—and something else which I shall call math or school math.
我认为“学校数学”是一种社会建构，一种 QWERTY 键盘。一系列历史事件（稍后将讨论）决定了某些数学主题的选择，作为公民应该承担的数学包袱。就像 QWERTY 键盘的打字机键排列一样，学校数学在特定的历史背景下确实有一定意义。但是，就像 QWERTY 键盘一样，它已经根深蒂固，以至于人们认为它是理所当然的，并在使它有意义的历史条件消失很久之后，还为它发明了合理化解释。事实上，对于我们文化中的大多数人来说，学校数学可能会有很大的不同，这是不可想象的：这是他们唯一知道的数学。为了打破这种恶性循环，我将带领读者进入一个新的数学领域，即海龟几何，这是我和同事为儿童创建的更好、更有意义的第一个正式数学领域。通过更仔细地观察造成学校数学形态的历史条件，可以最好地理解海龟几何的设计标准。
I see “school math” as a social construction, a kind of qwerty. A set of historical accidents (which shall be discussed in a moment) determined the choice of certain mathematical topics as the mathematical baggage that citizens should carry. Like the qwerty arrangement of typewriter keys, school math did make some sense in a certain historical context. But, like qwerty, it has dug itself in so well that people take it for granted and invent rationalizations for it long after the demise of the historical conditions that made sense of it. Indeed, for most people in our culture it is inconceivable that school math could be very much different: This is the only mathematics they know. In order to break this vicious circle I shall lead the reader into a new area of mathematics, Turtle geometry, that my colleagues and I have created as a better, more meaningful first area of formal mathematics for children. The design criteria of Turtle geometry are best understood by looking a little more closely at the historical conditions responsible for the shape of school math.
其中一些历史条件是务实的。在电子计算器出现之前，许多人被“编程”以快速准确地执行长除法等运算是一种实际的社会需要。但是，既然我们可以廉价地购买计算器，我们就应该重新考虑是否有必要让每个孩子花费数百个小时来学习这些算术函数。我并不是要否认一些关于数字的知识，事实上，很多关于数字的知识的智力价值。远非如此。但我们现在可以根据连贯、理性的理由来选择这些知识。我们可以摆脱肤浅、务实的考虑的束缚，这些考虑决定了过去关于应该学习什么知识以及在什么年龄学习的选择。
Some of these historical conditions were pragmatic. Before electronic calculators existed, it was a practical social necessity that many people be “programmed” to perform such operations as long division quickly and accurately. But now that we can purchase calculators cheaply we should reconsider the need to expend several hundred hours of every child’s life on learning such arithmetic functions. I do not mean to deny the intellectual value of some knowledge, indeed, of a lot of knowledge, about numbers. Far from it. But we can now select this knowledge on coherent, rational grounds. We can free ourselves from the tyranny of the superficial, pragmatic considerations that dictated past choices about what knowledge should be learned and at what age.
但实用性只是学校数学的历史原因之一。其他原因则与数学有关。数学是一套指导学习的指导原则。学校数学的一些历史原因与前计算机时代可学可教的内容有关。在我看来，决定学校数学中数学内容的一个主要因素是，在学校教室中，用原始的铅笔和纸技术可以做什么。例如，孩子们可以用铅笔和纸画图。所以决定让孩子们画很多图。同样的考虑也影响了对某些几何类型的强调。例如，在学校数学中，“解析几何”已成为用方程表示曲线的同义词。因此，每个受过教育的人都模糊地记得y = x 2是抛物线方程。尽管大多数父母几乎不知道为什么有人应该知道这一点，但当他们的孩子不知道时，他们会感到愤慨。他们认为，那些更了解这些事情的人一定知道一个深刻而客观的原因。讽刺的是，数学恐惧症使大多数人不敢更深入地研究这些原因，因此只能任由自封的数学专家摆布。很少有人会想到，学校数学中包括和不包括的内容可能与用铅笔画抛物线的简易性一样粗糙，技术含量很高！在计算机丰富的世界中，这可能会带来最深刻的变化：易于生成的数学结构的范围将大大扩展。
But utility was only one of the historical reasons for school math. Others were of a mathetic nature. Mathetics is the set of guiding principles that govern learning. Some of the historical reasons for school math had to do with what was learnable and teachable in the precomputer epoch. As I see it, a major factor that determined what mathematics went into school math was what could be done in the setting of school classrooms with the primitive technology of pencil and paper. For example, children can draw graphs with pencil and paper. So it was decided to let children draw many graphs. The same considerations influenced the emphasis on certain kinds of geometry. For example, in school math “analytic geometry” has become synonymous with the representation of curves by equations. As a result, every educated person vaguely remembers that y = x2 is the equation of a parabola. And although most parents have very little idea of why anyone should know this, they become indignant when their children do not. They assume that there must be a profound and objective reason known to those who better understand these things. Ironically, their mathophobia keeps most people from trying to examine those reasons more deeply and thus places them at the mercy of the self-appointed math specialists. Very few people ever suspect that the reason for what is included and what is not included in school math might be as crudely technological as the ease of production of parabolas with pencils! This is what could change most profoundly in a computer-rich world: The range of easily produced mathematical constructs will be vastly expanded.
学校数学社会建构的另一个数学因素是评分技术。活语言是通过口语来学习的，不需要老师来验证和评分每个句子。死语言需要老师不断的“反馈”。在学校数学中，被称为“求和”的活动发挥了这种反馈功能。这些荒谬的重复性小练习只有一个优点：它们很容易评分。但正是这个优点让它们在学校数学中占据了中心地位。简而言之，我认为学校数学的建构受到数学作为“死”科目时似乎可以教授的东西的强烈影响，当时使用棍子和沙子、粉笔和黑板、铅笔和纸等原始、被动的技术。结果是一组智力上不连贯的主题，违反了使某些材料容易学习而某些材料几乎不可能学习的最基本的数学原理。
Another mathetic factor in the social construction of school math is the technology of grading. A living language is learned by speaking and does not need a teacher to verify and grade each sentence. A dead language requires constant “feedback” from a teacher. The activity known as “sums” performs this feedback function in school math. These absurd little repetitive exercises have only one merit: They are easy to grade. But this merit has bought them a firm place at the center of school math. In brief, I maintain that construction of school math is strongly influenced by what seemed to be teachable when math was taught as a “dead” subject, using the primitive, passive technologies of sticks and sand, chalk and blackboard, pencil and paper. The result was an intellectually incoherent set of topics that violates the most elementary mathetic principles of what makes certain material easy to learn and some almost impossible.
面对学校的传统，数学教育可以采取两种方法。传统方法将学校数学视为既定实体，并努力寻找教授它的方法。一些教育工作者为此目的使用计算机。因此，自相矛盾的是，计算机在教育中最常见的用途是强行灌输前计算机时代遗留下来的难以消化的材料。在海龟几何中，计算机有完全不同的用途。在那里，计算机被用作数学表达媒介，它让我们可以自由地为儿童设计具有个人意义、智力连贯且易于学习的数学主题。我们不是将教育问题提出为“如何教授现有的学校数学”，而是将其提出为“重建数学”，或者更笼统地说，重建知识，以便不需要付出很大努力就可以教授它。
Faced with the heritage of school, math education can take two approaches. The traditional approach accepts school math as a given entity and struggles to find ways to teach it. Some educators use computers for this purpose. Thus, paradoxically, the most common use of the computer in education has become force-feeding indigestible material left over from the precomputer epoch. In Turtle geometry the computer has a totally different use. There the computer is used as a mathematically expressive medium, one that frees us to design personally meaningful and intellectually coherent and easily learnable mathematical topics for children. Instead of posing the educational problem as “how to teach the existing school math,” we pose it as “reconstructing mathematics,” or more generally, as reconstructing knowledge in such a way that no great effort is needed to teach it.
所有的“课程发展”都可以被描述为“知识重构”。例如，20 世纪 60 年代的新数学课程改革曾试图改变学校数学的内容。但进展不大。它只能做加法，尽管是不同的加法。新加法处理的是集合而不是数字，或者以 2 进制而不是 10 进制进行算术运算，但这些事实并没有多大区别。此外，数学改革并没有对富有创造力的数学家的创造力提出挑战，因此从未获得标志着新思想产物的兴奋之光。“新数学”这个名字本身就是一个误称。它的数学内容几乎没有什么新意：它不是来自儿童数学的发明过程，而是来自数学家数学的琐碎化过程。孩子们需要也应该得到比挑选旧数学更好的东西。就像传给弟弟妹妹的衣服一样，它永远不会合身。
All “curriculum development” could be described as “reconstructing knowledge.” For example, the New Math curriculum reform of the sixties made some attempt to change the content of school math. But it could not go very far. It was stuck with having to do sums, albeit different sums. The fact that the new sums dealt with sets instead of numbers, or arithmetic in base two instead of base ten made little difference. Moreover, the math reform did not provide a challenge to the inventiveness of creative mathematicians and so never acquired the sparkle of excitement that marks the product of new thought. The name itself—“New Math”—was a misnomer. There was very little new about its mathematical content: It did not come from a process of invention of children’s mathematics but from a process of trivialization of mathematician’s mathematics. Children need and deserve something better than selecting out pieces of old mathematics. Like clothing passed down to the younger siblings, it never fits comfortably.
龟几何的最初目标是适合儿童。其主要设计标准是可应用性。当然，它必须具有严肃的数学内容，但我们将会看到，可应用性和严肃的数学思维并不矛盾。相反，我们最终会明白，一些最个人化的知识也是最深刻的数学知识。在许多方面，数学——例如空间和运动的数学以及重复的动作模式——对儿童来说是最自然的。正是在这些数学中，我们扎下了龟几何的主根。当我和同事们研究这些想法时，许多原则为可应用数学的概念提供了更多的结构。首先是连续性原则：数学必须与成熟的个人知识相连续，从中它可以继承一种温暖和价值感以及“认知”能力。然后是力量原则：它必须赋予学习者权力，使他们能够完成没有它就无法完成的有意义的个人项目。最后，还有一条文化共鸣原则：这个话题必须在更大的社会背景下有意义。我曾说过，海龟几何对孩子有意义。但除非成年人也能接受，否则它对孩子来说不会真正有意义。我们不能允许自己把有尊严的数学强加给孩子，就像我们自己没有理由吃不愉快的药一样。
Turtle geometry started with the goal of fitting children. Its primary design criterion was to be appropriable. Of course it had to have serious mathematical content, but we shall see that appropriability and serious mathematic thinking are not at all incompatible. On the contrary: We shall end up understanding that some of the most personal knowledge is also the most profoundly mathematical. In many ways mathematics—for example the mathematics of space and movement and repetitive patterns of action—is what comes most naturally to children. It is into this mathematics that we sink the taproot of Turtle geometry. As my colleagues and I have worked through these ideas, a number of principles have given more structure to the concept of an appropriable mathematics. First, there was the continuity principle: The mathematics must be continuous with well-established personal knowledge from which it can inherit a sense of warmth and value as well as “cognitive” competence. Then there was the power principle: It must empower the learner to perform personally meaningful projects that could not be done without it. Finally there was a principle of cultural resonance: The topic must make sense in terms of a larger social context. I have spoken of Turtle geometry making sense to children. But it will not truly make sense to children unless it is accepted by adults too. A dignified mathematics for children cannot be something we permit ourselves to inflict on children, like unpleasant medicine, although we see no reason to take it ourselves.
脚注
Footnotes
I原意是“博学多识”这个词，指博学多识的人。一个不太为人熟知的词，词干相同，我将在后面的章节中使用，即“数学”，与学习有关。
I The original meaning is present in the word “polymath,” a person of many learnings. A less well-known word with the same stem which I shall use in later chapters is “mathetic,” having to do with learning.
II人们与孩子一起生活了很长时间。我们不得不等待皮亚杰告诉我们孩子是如何思考的，以及我们忘记了童年时的思维方式，这一事实是如此引人注目，以至于它暗示了弗洛伊德的“认知压抑”模型。
II People have lived with children for a long time. The fact that we had to wait for Piaget to tell us how children think and what we all forget about our thinking as children is so remarkable that it suggests a Freudian model of “cognitive repression.”
第三章
CHAPTER 3
龟几何
Turtle Geometry
为学习而生的数学
A Mathematics Made for Learning
龟几何是一种不同的几何风格，就像欧几里得的公理风格和笛卡尔的解析风格彼此不同一样。欧几里得的几何风格是逻辑风格。笛卡尔的几何风格是代数风格。龟几何是一种计算风格的几何。
TURTLE GEOMETRY IS A DIFFERENT STYLE OF DOING GEOMETRY, JUST as Euclid’s axiomatic style and Descartes’s analytic style are different from one another. Euclid’s is a logical style. Descartes’s is an algebraic style. Turtle geometry is a computational style of geometry.
欧几里得从一组基本概念中建立了他的几何学，其中之一就是点。点可以定义为具有位置但没有其他属性的实体——它没有颜色、没有大小、没有形状。尚未进入正规数学领域、尚未“数学化”的人常常发现这个概念难以理解，甚至很奇怪。他们很难将它与他们所知道的任何其他事物联系起来。乌龟几何也有一个类似于欧几里得点的基本实体。但是这个我称之为“乌龟”的实体可以与人们所知道的事物联系起来，因为与欧几里得的点不同，它没有完全被剥夺所有属性，而且它不是静态的，而是动态的。除了位置之外，乌龟还有另一个重要属性：它有“方向”。欧几里得点位于某个位置——它有一个位置，这就是你对它的全部描述。乌龟位于某个位置——它也有位置——但它也面向某个方向——它的航向。在这方面，乌龟就像一个人——我在这里，面朝北方——或者一只动物或一条船。正是这些相似之处让乌龟具有了特殊的能力，可以作为孩子们第一个了解形式数学的代表。孩子们可以与乌龟产生共鸣，从而能够将他们对身体及其运动方式的了解运用到学习形式几何的工作中。
Euclid built his geometry from a set of fundamental concepts, one of which is the point. A point can be defined as an entity that has a position but no other properties—it has no color, no size, no shape. People who have not yet been initiated into formal mathematics, who have not yet been “mathematized,” often find this notion difficult to grasp, and even bizarre. It is hard for them to relate it to anything else they know. Turtle geometry, too, has a fundamental entity similar to Euclid’s point. But this entity, which I call a “Turtle,” can be related to things people know because unlike Euclid’s point, it is not stripped so totally of all properties, and instead of being static it is dynamic. Besides position the Turtle has one other important property: It has “heading.” A Euclidean point is at some place—it has a position, and that is all you can say about it. A Turtle is at some place—it, too, has a position—but it also faces some direction—its heading. In this, the Turtle is like a person—I am here and I am facing north—or an animal or a boat. And from these similarities comes the Turtle’s special ability to serve as a first representative of formal mathematics for a child. Children can identify with the Turtle and are thus able to bring their knowledge about their bodies and how they move into the work of learning formal geometry.
要了解这是如何发生的，我们需要了解有关 Turtles 的另一件事：它们能够接受用一种名为TURTLE TALK的语言表达的命令。FORWARD 命令使 Turtle 沿其面对的方向沿直线移动（见图3）。要告诉它要走多远，FORWARD后面必须跟一个数字：FORWARD 1 将导致非常小的移动，FORWARD 100 将导致更大的移动。在LOGO环境中，许多孩子通过向他们介绍机械海龟（一种控制论机器人）开始学习 Turtle 几何，当这些命令在打字机键盘上输入时，它会执行这些命令。这个“地板海龟”有轮子、圆顶形状和一支笔，因此它可以在移动时画一条线。但它的基本属性（位置、航向和服从TURTLE TALK命令的能力）才是进行几何学研究的关键。孩子们以后可能会在 Turtle 的另一个化身“光海龟”中遇到这三个相同的属性。这是电视屏幕上的一个三角形物体。它也有位置和方向。它也会根据相同的TURTLE TALK命令移动。每种 Turtle 都有各自的优点：地板 Turtle 既可以用作推土机，也可以用作绘图工具；Light Turtle 可以绘制色彩鲜艳的线条，速度比眼睛跟上的速度还快。没有哪一种更好，但有两个 Turtle 这一事实蕴含着一个强大的理念：两个物理上不同的实体在数学上可以是相同的（或“同构”）。1
To see how this happens we need to know one more thing about Turtles: They are able to accept commands expressed in a language called TURTLE TALK. The command FORWARD causes the Turtle to move in a straight line in the direction it is facing (see Figure 3). To tell it how far to go, FORWARD must be followed by a number: FORWARD 1 will cause a very small movement, FORWARD 100 a larger one. In LOGO environments many children have been started on the road to Turtle geometry by introducing them to a mechanical turtle, a cybernetic robot, that will carry out these commands when they are typed on a typewriter keyboard. This “floor Turtle” has wheels, a dome shape, and a pen so that it can draw a line as it moves. But its essential properties—position, heading, and ability to obey TURTLE TALK commands—are the ones that matter for doing geometry. The child may later meet these same three properties in another embodiment of the Turtle: a “Light Turtle.” This is a triangular-shaped object on a television screen. It too has a position and a heading. And it too moves in response to the same TURTLE TALK commands. Each kind of Turtle has its strong points: The floor Turtle can be used as a bulldozer as well as a drawing instrument; the Light Turtle draws bright-colored lines faster than the eye can follow. Neither is better, but the fact that there are two carries a powerful idea: Two physically different entities can be mathematically the same (or “isomorphic”).1
FORWARD和BACK命令使 Turtle 沿其前进方向直线移动：其位置会改变，但其前进方向保持不变。另外两个命令会改变前进方向而不影响位置：RIGHT和LEFT使 Turtle “转动”，即在保持原位的同时改变前进方向。与FORWARD一样，转弯命令也需要给定一个数字（输入消息），以指示 Turtle 应转弯的程度。成人会很快识别出这些数字是转弯角度的度量单位。对于大多数儿童来说，这些数字必须得到探索，而探索过程既令人兴奋又充满乐趣。
The commands FORWARD and BACK cause a Turtle to move in a straight line in the direction of its heading: Its position changes, but its heading remains the same. Two other commands change the heading without affecting the position: RIGHT and LEFT cause a Turtle to “pivot,” to change heading while remaining in the same place. Like FORWARD, a turning command also needs to be given a number—an input message—to say how much the Turtle should turn. An adult will quickly recognize these numbers as the measure of the turning angle in degrees. For most children these numbers have to be explored, and doing so is an exciting and playful process.
可以通过命令生成正方形
A square can be produced by the commands
前进100
FORWARD 100
右 90
RIGHT 90
前进100
FORWARD 100
右 90
RIGHT 90
前进100
FORWARD 100
右 90
RIGHT 90
前进100
FORWARD 100
右 90
RIGHT 90
FD 100（注意缩写以减少打字）
FD 100 (note the abbreviations to reduce typing)
RT 100
RT 100
FD 100
FD 100
ERASE 1（这将撤消上一个命令的效果）
ERASE 1 (this undoes the effect of the previous command)
RT10（摆弄乌龟寻找正确的角度）
RT10 (fiddling the turtle in search of the right angle)
LT 10
LT 10
LT 10
LT 10
FD 100
FD 100
RT 100（这次到达得更快）
RT 100 (gets there faster this time)
LT 10
LT 10
RT 100
RT 100
LT 10
LT 10
FD 100
FD 100
RT 40
RT 40
FD 100
FD 100
RT 90
RT 90
FD 100
FD 100
图 3. 孩子早期尝试玩方块游戏的真实记录
Figure 3. An Actual Transcript of a Child’s Early Attempt at a Square
因为学习控制乌龟就像学习说一门语言，所以它调动了孩子说话的专业知识和乐趣。因为它就像在发号施令，所以它调动了孩子指挥的专业知识和乐趣。要让乌龟画出一个正方形，你自己要走进一个正方形，并在乌龟 谈话中描述你在做什么。所以，与乌龟一起工作可以调动孩子的专业知识和运动的乐趣。它利用孩子对“身体几何”的既定知识作为发展形式几何的桥梁的起点。
Since learning to control the Turtle is like learning to speak a language, it mobilizes the child’s expertise and pleasure in speaking. Since it is like being in command, it mobilizes the child’s expertise and pleasure in commanding. To make the Turtle trace a square, you walk in a square yourself and describe what you are doing in TURTLE TALK. And so, working with the Turtle mobilizes the child’s expertise and pleasure in motion. It draws on the child’s well-established knowledge of “body-geometry” as a starting point for the development of bridges into formal geometry.
儿童在 Turtle 学习环境中的首次体验的目标不是学习正式规则，而是培养对空间中移动方式的洞察力。这些洞察力在TURTLE TALK中进行了描述，并因此成为 Turtle 的“程序”或“过程”或“微分方程”。让我们仔细看看一个已经学会沿直线移动 Turtle 来绘制正方形、三角形和矩形的孩子如何学习如何编程来绘制圆形。
The goal of children’s first experiences in the Turtle learning environment is not to learn formal rules but to develop insights into the way they move about in space. These insights are described in TURTLE TALK and thereby become “programs” or “procedures” or “differential equations” for the Turtle. Let’s look closely at how a child, who has already learned to move the Turtle in straight lines to draw squares, triangles, and rectangles, might learn how to program it to draw a circle.
那么，让我们想象一下，就像我见过无数次的那样，一个孩子问：我怎样才能让乌龟画一个圆圈？LOGO环境中的指导员不会提供这类问题的答案，而是向孩子介绍一种不仅能解决这个问题而且能解决一大堆其他问题的方法。这种方法可以用一句话来概括：“玩乌龟”。鼓励孩子移动自己的身体，因为屏幕上的乌龟必须移动才能画出所需的图案。对于想要画一个圆圈的孩子来说，在圆圈中移动可能会得到这样的描述：“当你在圆圈中行走时，你向前迈出一小步，然后稍微转一圈。一直这样做。”从这个描述到正式的乌龟程序只有一小步了。
Let us imagine, then, as I have seen a hundred times, a child who demands: How can I make the Turtle draw a circle? The instructor in a LOGO environment does not provide answers to such questions but rather introduces the child to a method for solving not only this problem but a large class of others as well. This method is summed up in the phrase “play Turtle.” The child is encouraged to move his or her body as the Turtle on the screen must move in order to make the desired pattern. For the child who wanted to make a circle, moving in a circle might lead to a description such as: “When you walk in a circle you take a little step forward and you turn a little. And you keep doing it.” From this description it is only a small step to a formal Turtle program.
循环重复 [向前 1 向右 1]
TO CIRCLE REPEAT [FORWARD 1 RIGHT 1]
另一个孩子可能缺乏简单的编程经验，也不熟悉“玩乌龟”的启发式方法，因此可能需要帮助。但这种帮助主要不是教孩子如何编写乌龟圈，而是教孩子一种方法，一种启发式程序。这种方法（包括总结为“玩乌龟”的建议）试图在个人活动和正式知识的创造之间建立牢固的联系。
Another child, perhaps less experienced in simple programming and in the heuristics of “playing Turtle,” might need help. But the help would not consist primarily of teaching the child how to program the Turtle circle, but rather of teaching the child a method, a heuristic procedure. This method (which includes the advice summed up as “play Turtle”) tries to establish a firm connection between personal activity and the creation of formal knowledge.
在 Turtle Mathland 中，拟人化图像有助于将知识从熟悉的环境转移到新环境。例如，通常所说的“编程计算机”的隐喻就是教 Turtle 一个新单词。如果孩子想画很多正方形，可以教 Turtle 一个新命令，让它按顺序执行画正方形的七个命令，如图3所示。这可以以几种不同的形式提供给计算机，其中包括：
In the Turtle Mathland, anthropomorphic images facilitate the transfer of knowledge from familiar settings to new contexts. For example, the metaphor for what is usually called “programming computers” is teaching the Turtle a new word. A child who wishes to draw many squares can teach the Turtle a new command that will make it carry out in sequence the seven commands used to draw a square as is shown in Figure 3. This can be given to the computer in several different forms, among which are:
平方
TO SQUARE
前进100
FORWARD 100
右 90
RIGHT 90
前进100
FORWARD 100
右 90
RIGHT 90
前进100
FORWARD 100
右 90
RIGHT 90
前进100
FORWARD 100
结尾
END
平方
TO SQUARE
重复4
REPEAT 4
前进100
FORWARD 100
右 90
RIGHT 90
结尾
END
到正方形：尺寸
TO SQUARE:SIZE
重复4
REPEAT 4
前锋：尺寸
FORWARD:SIZE
右 90
RIGHT 90
结尾
END
类似地，我们可以通过以下方式绘制等边三角形：
Similarly we can program an equilateral triangle by:
三角
TO TRIANGLE
前进100
FORWARD 100
右 120
RIGHT 120
前进100
FORWARD 100
右 120
RIGHT 120
前进100
FORWARD 100
结尾
END
至三角形：侧面
TO TRIANGLE:SIDE
重复3
REPEAT 3
前锋：侧锋
FORWARD:SIDE
右 1200
RIGHT 1200
结尾
END
这些替代程序几乎达到了相同的效果，但是熟悉的读者会注意到一些差异。最明显的区别是，有些程序允许绘制不同大小的图形：在这种情况下，绘制图形的命令必须是SQUARE 50 或SQUARE 100 ，而不是简单的SQUARE。更微妙的区别是，有些程序让 Turtle 保持其原始状态。用这种简洁风格编写的程序更容易理解，并且在各种情况下都更容易使用。注意到这种差异，孩子们会学到两种教训。他们学习一个一般的“数学原理”，制造有利于模块化的组件。他们还学会使用非常强大的“状态”概念。
These alternative programs achieve almost the same effects, but informed readers will notice some differences. The most obvious difference is in the fact that some of them allow figures to be drawn with different sizes: In these cases the command to draw the figure would have to be SQUARE 50 or SQUARE 100 rather than simply SQUARE. A more subtle difference is in the fact that some of them leave the Turtle in its original state. Programs written in this clean style are much easier to understand and use in a variety of contexts. And in noticing this difference, children learn two kinds of lessons. They learn a general “mathetic principle,” making components to favor modularity. And they learn to use the very powerful idea of “state.”
同样的从熟悉到未知的策略可以让学习者接触到一些强有力的一般思想：例如，层次组织的思想（知识、组织和有机体的思想）、实施项目的规划思想，以及调试的思想。
The same strategy of moving from the familiar to the unknown brings the learner into touch with some powerful general ideas: for example, the idea of hierarchical organization (of knowledge, of organizations, and of organisms), the idea of planning in carrying through a project, and the idea of debugging.
画三角形或正方形并不需要计算机。铅笔和纸就可以了。但是一旦这些程序被构建出来，它们就变成了积木，使孩子能够创建知识的层次结构。在这个过程中，强大的智力技能得到了发展——这一点在孩子们和 Turtle 进行几次课程后为自己设定的一些项目中最明显地体现出来。许多孩子自发地走上了与 Pamela 相同的道路。她从教计算机SQUARE和TRIANGLE开始，正如前面所述。现在她发现她可以通过将三角形放在正方形上来建造一所房子。所以她尝试：
One does not need a computer to draw a triangle or a square. Pencil and paper would do. But once these programs have been constructed they become building blocks that enable a child to create hierarchies of knowledge. Powerful intellectual skills are developed in the process—a point that is most clearly made by looking at some projects children have set for themselves after a few sessions with the Turtle. Many children have spontaneously followed the same path as Pamela. She began by teaching the computer SQUARE and TRIANGLE as described previously. Now she saw that she can build a house by putting the triangle on top of the square. So she tries:
到家
TO HOUSE
正方形
SQUARE
三角形
TRIANGLE
结尾
END
但是当她发出HOUSE命令时，海龟画出了（图 4）：
But when she gives the command HOUSE, the Turtle draws (Figure 4):
图 4
Figure 4
三角形出现在正方形内部，而不是在正方形上方！
The triangle came out inside the square instead of on top of it!
通常在数学课上，孩子对错误答案的反应是试图尽快忘记它。但在LOGO环境中，孩子不会因为绘图错误而受到批评。调试过程是理解程序过程的正常部分。鼓励程序员研究错误而不是忘记错误。而在 Turtle 环境中，研究错误是有充分理由的。它会有所回报。
Typically in math class, a child’s reaction to a wrong answer is to try to forget it as fast as possible. But in the LOGO environment, the child is not criticized for an error in drawing. The process of debugging is a normal part of the process of understanding a program. The programmer is encouraged to study the bug rather than forget the error. And in the Turtle context there is a good reason to study the bug. It will pay off.
图 5
Figure 5
修复这个 bug 的方法有很多。Pamela 在玩 Turtle 游戏时发现了其中一种方法。她沿着 Turtle 的轨迹行走，发现三角形进入了正方形内部，因为三角形开始的第一个旋转动作是右转。因此，她可以通过编写左转三角形程序来修复这个 bug。修复这个 bug 的另一种常用方法是在SQUARE和 TRIANGLE之间插入一个RIGHT 30。无论哪种情况，修改后的程序都会生成以下图片（图 5）。
There are many ways this bug can be fixed. Pamela found one of them by playing Turtle. By walking along the Turtle’s track she saw that the triangle got inside the square because its first turning move in starting the triangle was a right turn. So she could fix the bug by making a left-turning triangle program. Another common way to fix this bug is by inserting a RIGHT 30 between SQUARE and TRIANGLE. In either case the amended procedure makes the following picture (Figure 5).
学习者看到了进步，也看到了事情并不总是完全正确或完全错误，而是处于一个连续体中。房子变好了，但仍然有一个缺陷。再玩一会儿 Turtle，这个最后的缺陷就被确定下来了，并通过在程序的第一步中向右转90 度来修复。
The learner sees progress, and also sees that things are not often either completely right or completely wrong but, rather, are on a continuum. The house is better but still has a bug. With a little more playing Turtle this final bug is pinned down and fixed by doing a RIGHT 90 as the first step in the program.
有些孩子使用程序构建块来绘制具体的图画，例如HOUSE。其他孩子则更喜欢抽象效果。例如，如果您发出命令SQUARE，将 Turtle 以RIGHT 120 旋转，再次执行SQUARE，将TURTLE以RT 120 或RT 10 旋转，再次执行SQUARE并不断重复，您将得到图 6a中的图片。较小的旋转将得到图 6b中的图片。
Some children use program building blocks to make concrete drawings such as HOUSE. Others prefer more abstract effects. For example, if you give the command SQUARE, pivot the Turtle with a RIGHT 120, do SQUARE again, pivot the TURTLE with RT 120 or with RT 10, do SQUARE once more and keep repeating, you get the picture in Figure 6a. A smaller rotation gives the picture in Figure 6b.
图 6a
Figure 6a
图 6b
Figure 6b
这些例子表明，连续性和幂原理使海龟几何变得易于学习。但我们希望它还能做点别的事情，打开智慧之门，最好是成为重要而有力思想的载体。即使在绘制这些简单的正方形和星星时，海龟也包含了一些重要思想：角度、受控重复、状态变化运算符。为了更系统地了解孩子们通过玩海龟学到了什么，我们首先要区分两种知识。一种是数学知识。海龟只是海龟几何这个庞大数学学科的一小部分，海龟几何是一种易于学习的几何，也是非常普遍的数学思想的有效载体。另一种知识是数学：关于学习的知识。首先，我们将更仔细地研究海龟体验的数学方面，然后再转向其更技术性的数学方面。当然，两者有重叠。
These examples show how the continuity and the power principles make Turtle geometry learnable. But we wanted it to do something else as well, to open intellectual doors, preferably to be a carrier of important, powerful ideas. Even in drawing these simple squares and stars the Turtle carried some important ideas: angle, controlled repetition, state-change operator. To give ourselves a more systematic overview of what children learn from working with the Turtle we begin by distinguishing between two kinds of knowledge. One kind is mathematical. The Turtles are only a small corner of a large mathematical subject, Turtle geometry, a kind of geometry that is easily learnable and an effective carrier of very general mathematical ideas. The other kind of knowledge is mathetic: knowledge about learning. First we shall look more closely at the mathetic aspects of the Turtle experience and then turn to its more technically mathematical side. Of course, the two overlap.
我们通过与基本数学原理联系起来介绍海龟几何：理解你想学的东西。回想一下珍妮的例子，她具备定义名词或动词的概念前提，但无法学习语法，因为她无法认同这一事业。从根本上讲，语法对她来说毫无意义。海龟几何是专门为儿童设计的，旨在成为一种可以理解的东西，能够与他们对重要事物的感受产生共鸣的东西。它旨在帮助儿童发展数学策略：为了学习某样东西，首先要理解它。
We introduced Turtle geometry by relating it to a fundamental mathetic principle: Make sense of what you want to learn. Recall the case of Jenny, who possessed the conceptual prerequisites for defining nouns or verbs but who could not learn grammar because she could not identify with this enterprise. In this very fundamental way, grammar did not make sense to her. Turtle geometry was specifically designed to be something children could make sense of, to be something that would resonate with their sense of what is important. And it was designed to help children develop the mathetic strategy: In order to learn something, first make sense of it.
乌龟圈事件说明了协同学习。2这个术语借用自临床心理学，可以与前面讨论过的分离学习形成对比。有时，该术语与指代各种协同性的限定词一起使用。例如，乌龟圈是身体协同的，因为这个圆圈与儿童对自己身体的感觉和知识紧密相关。或者它是自我协同的，因为它与儿童对自己作为有意图、目标、愿望、好恶的人的感觉相一致。画乌龟圈的孩子想要画这个圆圈；这样做会产生自豪感和兴奋感。
The Turtle circle incident illustrates syntonic learning.2 This term is borrowed from clinical psychology and can be contrasted to the dissociated learning already discussed. Sometimes the term is used with qualifiers that refer to kinds of syntonicity. For example, the Turtle circle is body syntonic in that the circle is firmly related to children’s sense and knowledge about their own bodies. Or it is ego syntonic in that it is coherent with children’s sense of themselves as people with intentions, goals, desires, likes, and dislikes. A child who draws a Turtle circle wants to draw the circle; doing it produces pride and excitement.
海龟几何是可以学习的，因为它是共振的。它还有助于学习其他东西，因为它鼓励自觉、刻意地使用解决问题和数学策略。数学家乔治·波利亚3认为应该教授解决问题的一般方法。海龟几何中使用的某些策略是波利亚建议的特例。例如，波利亚建议，每当我们着手解决问题时，都应该在脑海中浏览一份启发式问题清单，例如：这个问题可以细分为更简单的问题吗？这个问题是否与我已经知道如何解决的问题相关？海龟几何非常适合这种练习。找出如何让海龟画圆的关键是参考一个众所周知的问题——绕圈行走的问题。海龟几何提供了练习分解难题艺术的绝佳机会。例如，HOUSE是通过先制作SQUARE和TRIANGLE来制作的。总之，我认为海龟几何与波利亚原理非常契合，所以向学生讲解波利亚原理的最好方式就是让他们学习海龟几何，这样海龟几何就成为启发式策略一般思想的载体。
Turtle geometry is learnable because it is syntonic. And it is an aid to learning other things because it encourages the conscious, deliberate use of problem-solving and mathetic strategies. Mathematician George Polya3 has argued that general methods for solving problems should be taught. Some of the strategies used in Turtle geometry are special cases of Polya’s suggestions. For example, Polya recommends that whenever we approach a problem we should run through a mental checklist of heuristic questions such as: Can this problem be subdivided into simpler problems? Can this problem be related to a problem I already know how to solve? Turtle geometry lends itself to this exercise. The key to finding out how to make a Turtle draw a circle is to refer to a problem whose solution is known very well indeed—the problem of walking in a circle. Turtle geometry provides excellent opportunities to practice the art of splitting difficulties. For example, HOUSE was made by first making SQUARE and TRIANGLE. In short, I believe that Turtle geometry lends itself so well to Polya’s principles that the best way to explain Polya to students is to let them learn Turtle geometry. Thus, Turtle geometry serves as a carrier for the general ideas of a heuristic strategy.
由于波利亚的影响，人们经常建议数学教师除了关注内容外，还应特别关注启发式或“过程”。这种想法未能在教育系统中扎根，部分原因在于缺乏良好的情境，在这些情境中，孩子们可以接触并内化简单而令人信服的启发式知识模型。海龟几何不仅在此类情境中表现得特别丰富，还为波利亚的建议增添了新元素：要解决问题，先寻找你已经理解的类似问题。这个建议是抽象的；海龟几何将其变成了一个具体的程序性原则：玩海龟游戏。自己动手。在海龟游戏中，几乎有取之不尽的“类似情况”来源，因为我们借鉴了自己的行为和身体。所以，遇到麻烦时，我们可以玩海龟游戏。这让波利亚的建议变得切实可行。海龟几何成为通往波利亚的桥梁。与海龟游戏进行过广泛接触的孩子深深地相信“寻找类似问题”的价值，因为这个建议往往很有效果。这些成功带来了信心和技能，使他们能够学习如何在类似性不太明显的情形中应用该原则，例如在学校数学中遇到的大多数情形。学校数学虽然在算术内容方面比较初级，但对于波利亚的原则的运用来说却是一门相对高级的学科。
Because of Polya’s influence, it has often been suggested that mathematics teachers pay explicit attention to heuristics or “process” as well as to content. The failure of this idea to take root in the educational system can be explained partially by the paucity of good situations in which simple and compelling models of heuristic knowledge can be encountered and internalized by children. Turtle geometry is not only particularly rich in such situations, it also adds a new element to Polya’s advice: To solve a problem, look for something like it that you already understand. The advice is abstract; Turtle geometry turns it into a concrete, procedural principle: Play Turtle. Do it yourself. In Turtle work an almost inexhaustible source of “similar situations” is available because we draw on our own behavior, our own bodies. So, when in trouble, we can play Turtle. This brings Polya’s advice down to earth. Turtle geometry becomes a bridge to Polya. The child who has worked extensively with Turtles becomes deeply convinced of the value of “looking for something like it” because the advice has often paid off. From these successes comes the confidence and skill needed to learn how to apply the principle in situations, such as most of those encountered in school math, where similarities are less evident. School math, though elementary in terms of its arithmetic content, is a relatively advanced subject for the exercise of Polya’s principles.
算术不是启发式思维的入门领域，而乌龟几何则是一个很好的领域。通过自我和身体的共振特性，学习画乌龟的动作为孩子提供了一种学习模式，这种模式与五年级男孩比尔描述的在学校学习乘法表的分离模式截然不同：“你通过让大脑一片空白，然后一遍又一遍地重复，直到记住为止，来学习那种东西。”比尔花了相当多的时间“学习”他的乘法表。结果很差，事实上，糟糕的结果本身就说明了比尔在学习过程中对自己心理过程的准确报告。他之所以学不好，是因为他强迫自己脱离与材料的任何关系——或者更确切地说，他采用了最糟糕的关系，即分离，作为学习策略。他的老师认为他“记忆力差”，甚至讨论了脑损伤的可能性。但比尔对流行歌曲和民歌有着广泛的了解，他毫不费力地就记住了，也许是因为他太忙了，没有时间去想，让自己的脑子一片空白。
Arithmetic is a bad introductory domain for learning heuristic thinking. Turtle geometry is an excellent one. By its qualities of ego and body syntonicity, the act of learning to make the Turtle draw gives the child a model of learning that is very much different from the dissociated one a fifth-grade boy, Bill, described as the way to learn multiplication tables in school: “You learn stuff like that by making your mind a blank and saying it over and over until you know it.” Bill spent a considerable amount of time on “learning” his tables. The results were poor, and in fact, the poor results themselves speak for the accuracy of Bill’s reporting of his own mental processes in learning. He failed to learn because he forced himself out of any relationship to the material—or rather, he adopted the worst relationship, dissociation, as a strategy for learning. His teachers thought that he “had a poor memory” and had even discussed the possibility of brain damage. But Bill had extensive knowledge of popular and folk songs, which he had no difficulty remembering, perhaps because he was too busy to think about making his mind a blank.
目前关于大脑功能分离的理论可能认为比尔的“记忆力差”只与数字有关。但这个男孩却能轻松地复述数千条记录的参考编号、价格和日期。他“能”学到什么和“不能”学到什么之间的区别并不取决于知识的内容，而是取决于他与知识的关系。海龟几何学与节奏和运动以及日常生活中所需的导航知识的联系，让比利能够像对待歌曲而不是乘法表一样对待它。他的进步令人惊叹。通过海龟几何学，比利以前拒绝的数学知识可以进入他的知识世界。
Current theories about the separation of brain functions might suggest that Bill’s “poor memory” was specific to numbers. But the boy could easily recount reference numbers, prices, and dates for thousands of records. The difference between what he “could” and “could not” learn did not depend on the content of the knowledge but on his relationship to it. Turtle geometry, by virtue of its connection with rhythm and movement and the navigational knowledge needed in everyday life, allowed Billy to relate to it more as he did to songs than to multiplication tables. His progress was spectacular. Through Turtle geometry, mathematical knowledge Billy had previously rejected could enter his intellectual world.
现在我们从数学转向数学。当一个人学习海龟几何时，他学到了什么数学知识？为了讨论的目的，我们区分了三类数学知识，每类知识都受益于与海龟的合作。首先，有知识体系“学校数学”，它被明确地选为（在我看来，这很大程度上是历史的偶然）所有公民都应该掌握的基础数学的核心。第二，有知识体系（我称之为“原始数学”），虽然传统课程没有明确提到，但学校数学预先假定了它。其中一些知识具有一般的“社会”性质：例如，与我们为什么要学习数学以及我们如何理解数学有关的知识。这一类别中的其他知识是发生认识论引起教育工作者注意的底层结构：演绎原理，如传递性、守恒定律、分类的直观逻辑等。最后，还有第三类：学校数学既未包括也未预设的知识，但应该考虑纳入未来受过教育的公民的智力装备中。
Now we turn from mathetic to mathematical considerations. What mathematics does one learn when one learns Turtle geometry? For the purposes of this discussion we distinguish three classes of mathematical knowledge, each of which benefits from work with Turtles. First, there is the body of knowledge “school math” that has been explicitly selected (in my opinion largely by historical accident) as the core of basic mathematics that all citizens should possess. Second, there is a body of knowledge (let me call it “protomath”) that is presupposed by school math even though it is not explicitly mentioned in traditional curricula. Some of this knowledge is of a general “social” nature: for example, knowledge that bears on why we do mathematics at all and how we can make sense of math. Other knowledge in this category is the kind of underlying structure to which genetic epistemology has drawn the attention of educators: deductive principles such as transitivity, the conservations, the intuitive logic of classifications, and so on. Finally, there is a third category: knowledge that is neither included in nor presupposed by the school math but that ought to be considered for inclusion in the intellectual equipment of the educated citizen of the future.
我认为理解欧几里得几何、笛卡尔几何和微分几何系统之间的关系属于第三类。对于学生来说，画一个 Turtle 圆不仅仅是一种“常识性”的画圆方法。它让孩子接触到微积分的核心思想。这一事实可能对许多读者来说是看不见的，因为他们只在高中或大学课程中接触过微积分，在这些课程中，“微积分”等同于某些符号的形式操作。Turtle 圆事件中的孩子并没有学习微积分的形式主义，例如x n的导数是nx n_1，而是学习了它的用法和含义。事实上，Turtle 圆程序为传统上称为“微分方程”的另一种形式主义，是微分背后思想的强大载体。这就是为什么可以通过 Turtle 理解如此多的主题；Turtle 程序是微分方程的直观类比，微分方程的概念几乎在传统应用数学的每个例子中都能找到。
I think that understanding the relations among the Euclidean, the Cartesian, and the differential systems of geometry belongs to this third category. For a student, drawing a Turtle circle is more than a “common sense” way of drawing circles. It places the child in contact with a cluster of ideas that lie at the heart of the calculus. This fact may be invisible to many readers whose only encounter with calculus was a high school or college course where “calculus” was equated with certain formal manipulations of symbols. The child in the Turtle circle incident was not learning about the formalism of calculus, for example that the derivative of xn is nxn_1, but about its use and its meaning. In fact the Turtle circle program leads to an alternative formalism for what is traditionally called a “differential equation” and is a powerful carrier of the ideas behind the differential. This is why it is possible to understand so many topics through the Turtle; the Turtle program is an intuitive analog of the differential equation, a concept one finds in almost every example of traditional applied mathematics.
微分学的强大之处在于它能够通过生长尖端发生的事情来描述生长过程。这就是它成为牛顿理解行星运动的一个很好的工具的原因。在描绘轨道时，行星现在所在位置的局部条件决定了它下一步将去往何处。在我们给乌龟的指令“前进1，右转 1”中，我们只提到了乌龟现在所在位置和它即将到达的位置之间的差异。这就是指令微分的原因。这里没有提到路径本身之外的任何遥远空间部分。乌龟在沿 圆周运动时，就像从内部看到圆周一样，对远离圆周的任何事物都视而不见。这个特性非常重要，数学家们给它起了一个特殊的名字：乌龟几何是“内在的”。内在微分几何的精神可以通过观察几种思考曲线（比如圆）的方式看出。对于欧几里得来说，圆的定义特征是圆上的点与圆心之间的距离是恒定的，而圆心本身不是圆的一部分。在笛卡尔几何中，在这方面更像欧几里得几何而不是乌龟几何，点的位置取决于它们与外部某物的距离，也就是垂直坐标轴。直线和曲线由连接这些坐标的方程定义。因此，例如，圆被描述为：
Differential calculus derives much of its power from an ability to describe growth by what is happening at the growing tip. This is what made it such a good instrument for Newton’s attempts to understand the motion of the planets. As the orbit is traced out, it is the local conditions at the place where the planet now finds itself that determine where it will go next. In our instructions to the Turtle, FORWARD 1, RIGHT TURN 1, we referred only to the difference between where the Turtle is now and where it shall momentarily be. This is what makes the instructions differential. There is no reference in this to any distant part of space outside of the path itself. The Turtle sees the circle as it goes along, from within, as it were, and is blind to anything far away from it. This property is so important that mathematicians have a special name for it: Turtle geometry is “intrinsic.” The spirit of intrinsic differential geometry is seen by looking at several ways to think about a curve, say, the circle. For Euclid, the defining characteristic of a circle is the constant distance between points on the circle and a point, the center, that is not itself part of the circle. In Descartes’s geometry, in this respect more like Euclid’s than that of the Turtle, points are situated by their distance from something outside of them, that is to say the perpendicular coordinate axes. Lines and curves are defined by equations connecting these coordinates. So, for example, a circle is described as:
（x_a ）2+（y_b）2 = R2
(x_a)2 +(y_b)2 = R2
在 Turtle 几何中，圆的定义是 Turtle 不断重复以下动作：向前一点，转一点。这种重复意味着它绘制的曲线将具有“恒定曲率”，曲率表示给定向前运动时转动的程度。4
In Turtle geometry a circle is defined by the fact that the Turtle keeps repeating the act: FORWARD a little, TURN a little. This repetition means that the curve it draws will have “constant curvature,” where curvature means how much you turn for a given forward motion.4
乌龟几何属于一类几何，其属性在欧几里得或笛卡尔系统中是不存在的。这些是自牛顿以来发展起来的微分几何，它们使许多现代物理学成为可能。我们已经注意到，微分方程是物理学描述粒子或行星运动的形式主义。在第 5 章中，我们将更详细地讨论这一点，我们还将看到它是描述动物运动或经济演变的适当形式主义。我们将更清楚地理解，乌龟几何与儿童的经历以及物理学中最强大的成就都有联系，这并非巧合。因为儿童的运动定律虽然形式上不那么精确，但它们与行星绕太阳旋转的运动定律以及飞蛾绕烛光旋转的运动定律共享微分方程的数学结构。而乌龟几何只不过是这种数学结构定性核心的直观计算形式的重建。当我们在第 5 章中回顾这些思想时，我们将看到海龟几何如何为直观掌握微积分、物理学和数学建模在生物和社会科学中的应用打开大门。
Turtle geometry belongs to a family of geometries with properties not found in the Euclidean or Cartesian system. These are the differential geometries that have developed since Newton and have made possible much of modern physics. We have noted that the differential equation is the formalism through which physics has been able to describe the motion of a particle or a planet. In chapter 5, where we discuss this in more detail, we shall also see that it is the appropriate formalism to describe the motion of an animal or the evolution of an economy. And we shall come to understand more clearly that it is not by coincidence that Turtle geometry has links both to the experience of a child and to the most powerful achievements in physics. For the laws of motion of the child, though less precise in form, share the mathematical structure of the differential equation with the laws of motion of planets turning about the sun and with those of moths turning about a candle flame. And the Turtle is nothing more or less than a reconstruction in intuitive computational form of the qualitative core of this mathematical structure. When we return to these ideas in chapter 5, we shall see how Turtle geometry opens the door to an intuitive grasp of calculus, physics, and mathematical modeling as it is used in the biological and social sciences.
海龟几何对学校数学某些部分的影响主要是关系或情感上的：许多孩子来到LOGO实验室时，讨厌数字，认为它们是外来事物，离开时却喜欢上了它们。在其他情况下，海龟几何为大多数孩子觉得困难的复杂数学概念提供了特定的直观模型。使用数字来测量角度就是一个简单的例子。在海龟几何环境中，孩子们几乎无意识地掌握了这种能力。每个人——包括我们接触过的少数一年级学生和许多三年级学生——都从这种体验中获得了比大多数高中生更好的 45 度、10 度或 360 度的含义。因此，他们为所有许多正式主题——几何、三角学、制图等——做好了准备，其中角度概念起着核心作用。但他们也为其他方面做好了准备，即我们社会使用角度测量的一个方面，而学校数学系统地对此视而不见。
The effect of work with Turtle geometry on some components of school math is primarily relational or affective: Many children have come to the LOGO lab hating numbers as alien objects and have left loving them. In other cases work with the Turtle provides specific intuitive models for complex mathematical concepts most children find difficult. The use of numbers to measure angles is a simple example. In the Turtle context children pick this ability up almost unconsciously. Everyone—including the few first graders and many third graders we have worked with—emerges from the experience with a much better sense of what is meant by 45 degrees or 10 degrees or 360 degrees than the majority of high school students ever acquire. Thus, they are prepared for all the many formal topics—geometry, trigonometry, drafting, and so on—in which the concept of angle plays a central part. But they are prepared for something else as well, an aspect of the use of angular measure in our society to which the school math is systematically blind.
角度概念在当代美国人生活中最广泛的表现之一是航海。数以百万计的人驾驶船只或飞机或阅读地图。对于大多数人来说，这些生活活动与死板的学校数学完全脱节。我们强调了这样一个事实：使用海龟作为角度概念的隐喻载体，将其与身体几何学紧密联系起来。我们称之为身体共振。在这里，我们看到了一种文化共振：海龟将角度概念与航海联系起来，这种活动牢固而积极地扎根于许多儿童的课外文化中。随着计算机继续在世界范围内传播，海龟几何的文化共振将变得越来越强大。
One of the most widespread representations of the idea of angle in the lives of contemporary Americans is in navigation. Many millions navigate boats or airplanes or read maps. For most there is a total dissociation between these live activities and the dead school math. We have stressed the fact that using the Turtle as metaphorical carrier for the idea of angle connects it firmly to body geometry. We have called this body syntonicity. Here we see a cultural syntonicity: The Turtle connects the idea of angle to navigation, activity firmly and positively rooted in the extraschool culture of many children. And as computers continue to spread into the world, the cultural syntonicity of Turtle geometry will become more and more powerful.
海龟帮助人们理解的第二个关键数学概念是变量的概念：使用符号来命名未知实体的概念。为了了解海龟对此有何贡献，我们将海龟圆程序扩展为海龟螺旋程序（图 7）。
A second key mathematical concept whose understanding is facilitated by the Turtle is the idea of a variable: the idea of using a symbol to name an unknown entity. To see how Turtles contribute to this, we extend the program for Turtle circles into a program for Turtle spirals (Figure 7).
以螺旋线为例。与圆一样，它也可以按照以下方法制作：向前走一点，转动一点。两者的区别在于，圆“始终相同”，而螺旋线从中间向外移动时会变得更平坦，“弯曲度更小”。圆是一条曲率恒定的曲线。螺旋线的曲率随其外移动而减小。要沿螺旋线行走，可以迈出一步，然后转动，迈出一步，然后转动，每次转动少一点（或迈出多一点）。要将其转化为对海龟的指令，您需要某种方式来表达您正在处理变量的事实。原则上，您可以通过一个非常长的程序（参见图 8）来描述这一点，该程序将精确指定海龟每一步应该转动多少度。这很乏味。更好的方法是使用通过变量进行符号命名的概念，这是有史以来最强大的数学思想之一。
Look, for example, at the coil spiral. Like the circle, it too can be made according to the prescription: Go forward a little, turn a little. The difference between the two is that the circle is “the same all the way” while the spiral gets flatter, “less curvy,” as you move out from the middle. The circle is a curve of constant curvature. The curvature of the spiral decreases as it moves outward. To walk in a spiral, one could take a step, then turn, take a step, then turn, each time turning a little less (or stepping a little more). To translate this into instructions for the Turtle, you need some way to express the fact that you are dealing with a variable quantity. In principle you could describe this by a very long program (see Figure 8) that would specify precisely how much the Turtle should turn on each step. This is tedious. A better method uses the concept of symbolic naming through a variable, one of the most powerful mathematical ideas ever invented.
图 7
Figure 7
图 8. 如何不画螺旋
Figure 8. How NOT to Draw Spirals
在TURTLE TALK中，变量被作为一种交流方式。我们想对 Turtle 说的是“向前走一小步，然后转动一定量，但现在我不能告诉你要转动多少，因为每次转动的量都不一样。”要画出“螺旋”，我们要说“向前走一段距离，每次距离都不一样，然后转 90 度。”在数学语言中，表达这种意思的技巧是为“我不能告诉你的量”发明一个名称。名称可以是一个字母，例如X ，也可以是一个完整的单词，例如ANGLE OR DISTANCE 。（计算机文化对数学的一个小贡献是它习惯使用助记词而不是单个字母作为变量的名称。）为了使变量的概念发挥作用，TURTLE TALK允许创建一个“带有输入的过程”。这可以通过输入以下内容来完成：
In TURTLETALK, variables are presented as a means of communication. What we want to say to the Turtle is “go forward a little step, then turn a certain amount, but I can’t tell you now how much to turn because it will be different each time.” To draw the “squiral” we want to say “go forward a certain distance, which will be different each time, and then turn 90.” In mathematical language the trick for saying something like this is to invent a name for the “amount I can’t tell you.” The name could be a letter, such as X, or it could be a whole word, such as ANGLE OR DISTANCE. (One of the minor contributions of the computer culture to mathematics is its habit of using mnemonic words instead of single letters as names for variables.) To put the idea of variable to work, TURTLE TALK allows one to create a “procedure with an input.” This can be done by typing:
步距
TO STEP DISTANCE
前锋距离
FORWARD DISTANCE
右 90
RIGHT 90
结尾
END
命令STEP 100 将使 Turtle 前进 100 个单位，然后向右转 90 度。同样，命令 STEP 110 将使它前进 110 个单位，然后向右转 90 度。在LOGO环境中，我们鼓励儿童使用拟人化的比喻：命令STEP调用代理（“ STEP人”），其工作是向 Turtle 发出两个命令，即FORWARD命令和RIGHT命令。但是，如果不向代理提供消息（一个数字），代理就无法执行此任务，该数字将传递给“ FORWARD人”，后者会将其传递给 Turtle。
The command STEP 100 will make the Turtle go forward 100 units and then turn right 90 degrees. Similarly STEP 110 will make it go forward 110 units and then turn 90 degrees. In LOGO environments we encourage children to use an anthropomorphic metaphor: The command STEP invokes an agent (a “STEP man”) whose job is to issue two commands, a FORWARD command and a RIGHT command, to the Turtle. But this agent cannot perform this job without being given a message—a number that will be passed on to the “FORWARD man” who will pass it on to the Turtle.
STEP过程其实并不十分令人兴奋，但只要稍加改动就能变得令人兴奋。将其与SPI过程进行比较，除了多一行之外，其余完全相同：
The procedure STEP is not really very exciting, but a small change will make it so. Compare it with the procedure SPI, which is exactly the same except for having one extra line:
至 SPI 距离
TO SPI DISTANCE
前锋距离
FORWARD DISTANCE
右 90
RIGHT 90
SPI 距离 + 5
SPI DISTANCE + 5
结尾
END
命令SPI 100 调用SPI代理并向其提供输入消息 100。然后， SPI代理发出三个命令。第一个命令与STEP代理的第一个命令一样：告诉 Turtle 前进 100 个单位。第二个命令告诉 Turtle 右转。同样，这没有什么新东西。但第三个命令做了一些不同寻常的事情。这个命令是SPI 105。它的效果是什么？它告诉 Turtle 前进 105 个单位，告诉 Turtle 右转 90 度，然后发出命令SPI 110。因此，我们有一个称为“递归”的技巧，用于设置一个永无止境的过程，其初始步骤如下页中的图 9a 和 9b 所示。
The command SPI 100 invokes a SPI agent and gives it the input message 100. The SPI agent then issues three commands. The first is just like the first command of the STEP agent: Tell the Turtle to go forward 100 units. The second tells the Turtle to turn right. Again there is nothing new. But the third does something extraordinary. This command is SPI 105. What is its effect? It tells the Turtle to go forward 105 units, tells the Turtle to turn right 90, and then issues the command SPI 110. Thus we have a trick called “recursion” for setting up a never-ending process whose initial steps are shown in Figures 9a and 9b on the following pages.
在我向孩子们介绍的所有想法中，递归是特别能引起兴奋反应的一个想法。我认为这部分是因为永无止境的想法触及了每个孩子的幻想，部分是因为递归本身植根于流行文化。例如，有一个递归谜语：如果你有两个愿望，第二个是什么？（还有两个愿望。）还有一张带有标签图片的令人回味的图片。通过打开玩无限的丰富机会，SPI程序所代表的一系列想法让孩子接触到成为数学家的感觉。图 9b说明了数学体验的另一个方面，我们可以看到如何通过改变SPI程序中的角度来探索一个奇怪的数学现象。接近 90 度的角度会产生一个令人惊讶的突发现象：星系的臂像扭曲的螺旋线，实际上并没有被编程到程序中。它们令人震惊，并常常激发人们进行长期的探索，将数字和几何思维与美学交织在一起。
Of all the ideas I have introduced to children, recursion stands out as the one idea that is particularly able to evoke an excited response. I think this is partly because the idea of going on forever touches on every child’s fantasies and partly because recursion itself has roots in popular culture. For example, there is the recursion riddle: If you have two wishes what is the second? (Two more wishes.) And there is the evocative picture of a label with a picture of itself. By opening the rich opportunities of playing with infinity the cluster of ideas represented by the SPI procedure puts a child in touch with something of what it is like to be a mathematician. Another aspect of living a mathematical experience is illustrated by Figure 9b where we see how a curious mathematical phenomenon can be explored by varying the angle in the SPI procedure. Angles close to 90 produce a surprising emergent phenomenon: The arms of the galaxy like twisted squirals were not actually programmed into the procedure. They come as a shock and often motivate long explorations in which numerical and geometric thinking intertwines with aesthetics.
图 9a
Figure 9a
图 9b
Figure 9b
在LOGO环境中，新想法的获得往往是为了满足个人需要，做一些以前做不到的事情。在传统的学校环境中，初学者会在一些小问题中遇到变量的概念，例如：
In the LOGO environment new ideas are often acquired as a means of satisfying a personal need to do something one could not do before. In a traditional school setting, the beginning student encounters the notion of variable in little problems such as:
5 + X = 8.X 是多少？
5 + X = 8. What is X?
很少有孩子将此视为与个人相关的问题，更少有孩子将解决方法视为力量源泉。他们是对的。在他们的生活背景下，他们用它做不了什么。在LOGO中，情况大不相同。在这里，孩子有个人需求：画一个螺旋。在这种情况下，变量的概念是个人力量的源泉，这种力量可以做某件想做的事情，但如果没有这个想法就无法实现。当然，许多在传统环境中接触变量概念的孩子确实学会了有效地使用它。但它很少传达出一种“数学力量”的感觉，即使是数学上最优秀和最聪明的孩子也是如此。这就是在传统学校和LOGO环境中接触变量概念的最大对比点。在LOGO中，这个概念赋予孩子力量，孩子体验到数学如何使整个文化能够做到以前没有人能做到的事情。
Few children see this as a personally relevant problem, and even fewer experience the method of solution as a source of power. They are right. In the context of their lives, they can’t do much with it. In the LOGO encounter, the situation is very much different. Here the child has a personal need: to make a spiral. In this context the idea of a variable is a source of personal power, power to do something desired but inaccessible without this idea. Of course, many children who encounter the notion of variable in a traditional setting do learn to use it effectively. But it seldom conveys a sense of “mathpower,” not even to the mathematically best and brightest. And this is the point of greatest contrast between an encounter with the idea of variables in the traditional school and in the LOGO environment. In LOGO, the concept empowers the child, and the child experiences what it is like for mathematics to enable whole cultures to do what no one could do before.
如果用变量画螺旋线作为孤立的例子来“说明”“数学力量的概念”，那么它只有偶然的机会与少数孩子产生共鸣（就像齿轮与我产生共鸣一样）。但在海龟几何中，它并不是一个孤立的例子。它是所有数学知识的典型表现。可以说，数学力量成为了一种生活方式。力量感不仅与立即适用的方法有关，例如使用变量的角度测量，还与“定理”、“证明”、“启发式”或“解决问题的方法”等概念有关。在使用这些概念时，孩子正在发展谈论数学的方式。我们现在要讨论的就是这种数学表达能力的发展。
If the use of a variable to make a spiral were introduced as an isolated example to “illustrate” the “concept of mathpower,” it would have only a haphazard chance of connecting with a few children (as gears connected with me). But in Turtle geometry it is not an isolated example. It is typical of how all mathematical knowledge is encountered. Mathpower, one might say, becomes a way of life. The sense of power is associated not only with immediately applicable methods such as the use of angular measure of variables but also with such concepts as “theorem” or “proof” or “heuristic” or “problem-solving method.” In using these concepts, the child is developing ways to talk about mathematics. And it is to this development of mathematical articulateness we now turn.
假设一个孩子已经用 Turtle 画了一个正方形和一个圆形，现在想画一个三角形，三边都等于 100 个 Turtle 步。程序的形式可能是：
Consider a child who has already made the Turtle draw a square and a circle and would now like to draw a triangle with all three sides equal to 100 Turtle steps. The form of the program might be:
三角
TO TRIANGLE
重复3
REPEAT 3
前进100
FORWARD 100
正确的事情
RIGHT SOMETHING
结尾
END
但是为了让 Turtle 画出图形，孩子需要告诉它更多。我们称之为SOMETHING 的数量是什么？对于正方形，我们指示 Turtle 在每个顶点处旋转 90 度，因此正方形程序为：
But for the Turtle to draw the figure, the child needs to tell it more. What is the quantity we called SOMETHING? For the square we instructed the Turtle to turn 90 degrees at each vertex, so that the square program was:
平方
TO SQUARE
重复4
REPEAT 4
前进100
FORWARD 100
右 90
RIGHT 90
结尾
END
现在我们可以看到波利亚的原则“寻找相似之处”和乌龟几何的程序原理“扮演乌龟”是如何协同工作的。正方形和三角形中有什么相同之处？如果我们扮演乌龟并“规划”我们希望乌龟进行的行程，我们会注意到在两种情况下，我们的起点和终点都是相同的，并且面向相同的方向。也就是说，我们以开始时的状态结束。在这期间，我们完成了一次完整的转弯。这两种情况的不同之处在于我们的转弯是“分三次”还是“分四次”完成的。这个想法的数学内容既强大又简单。优先考虑的是整个行程的概念——你在整个过程中转了多少圈？
Now we can see how Polya’s precept, “find similarities,” and Turtle geometry’s procedural principle, “play Turtle,” can work together. What is the same in the square and the triangle? If we play Turtle and “pace out” the trip that we want the Turtle to take, we notice that in both cases we start and end at the same point and facing the same direction. That is, we end in the state in which we started. And in between we did one complete turn. What is different in the two cases is whether our turning was done “in three goes” or “in four goes.” The mathematical content of this idea is as powerful as it is simple. Priority goes to the notion of the total trip—how much do you turn all the way around?
令人惊奇的是，所有行程的总转角都是相同的 360 度。正方形的四个 90 度等于 360 度，由于所有转角都发生在角落，三角形中的三个转角必须分别为 360 度除以三。所以我们称之为“某个量”的量实际上是 120 度。这就是“乌龟总行程定理”的命题。
The amazing fact is that all total trips turn the same amount, 360 degrees. The four 90 degrees of the square make 360 degrees, and since all the turning happens at the corner, the three turns in a triangle must each be 360 degrees divided by three. So the quantity we called SOMETHING is actually 120 degrees. This is the proposition of “The Total Turtle Trip Theorem.”
如果一只乌龟绕着任何区域的边界走了一圈，最后回到了它开始的状态，那么所有转弯的总和就是 360 度。5
If a Turtle takes a trip around the boundary of any area and ends up in the state in which it started, then the sum of all turns will be 360 degrees.5
理解这一点的关键是学习一种使用它来解决一类明确定义的问题的方法。因此，孩子遇到这个定理与记住它的欧几里得对应定理有几点不同：“三角形内角和为 180 度。”
Part and parcel of understanding this is learning a method of using it to solve a well-defined class of problems. Thus the child’s encounter with this theorem is different in several ways from memorizing its Euclidean counterpart: “The sum of the internal angles of a triangle is 180 degrees.”
首先（至少在LOGO计算机的背景下），全海龟行程定理更强大：儿童实际上可以使用它。其次，它更通用：它适用于正方形和曲线以及三角形。第三，它更易理解：它的证明很容易掌握。而且它更个性化：你可以“走一遍”，它是将数学与个人知识联系起来的一般习惯的典范。
First (at least in the context of LOGO computers), the Total Turtle Trip Theorem is more powerful: The child can actually use it. Second, it is more general: It applies to squares and curves as well as to triangles. Third, it is more intelligible: Its proof is easy to grasp. And it is more personal: You can “walk it through,” and it is a model for the general habit of relating mathematics to personal knowledge.
我们已经看到孩子们使用“全乌龟行程定理”画出等边三角形。但令人兴奋的是看到这个定理如何陪伴他们从简单的项目走向更高级的项目——书中央复制的方框中的花朵展示了一个项目沿着这条路径前进的一小段路。因为当我们给孩子们一个定理时，重要的不是让他们记住它。最重要的是，通过在成长过程中掌握一些非常强大的定理，人们开始欣赏某些思想如何可以作为一生的思考工具。人们学会享受和尊重强大思想的力量。人们了解到最强大的思想就是强大思想的思想。
We have seen children use the Total Turtle Trip Theorem to draw an equilateral triangle. But what is exciting is to watch how the theorem can accompany them from such simple projects to far more advanced ones—the flowers in the boxes that are reproduced in the center of the book show a project a little way along this path. For what is important when we give children a theorem to use is not that they should memorize it. What matters most is that by growing up with a few very powerful theorems, one comes to appreciate how certain ideas can be used as tools to think with over a lifetime. One learns to enjoy and to respect the power of powerful ideas. One learns that the most powerful idea of all is the idea of powerful ideas.
接下来是两个孩子在玩电脑时的假设对话。这些实验和其他实验每天都会发生——而且确实如此。
What follows is a hypothetical conversation between two children who are working and playing with the computer. These and other experiments can happen every day—and they do.
…一个开始
… A BEGINNING
该项目的下一阶段将产生最壮观的效果，因为鸟儿开始运动。但印刷的纸张无法捕捉产品或过程：偶然的发现、错误和数学见解都需要运动才能被欣赏。反思你所忽略的东西让我想到了计算机为孩子提供的另一个新事物：在运动中绘画的机会，实际上是涂鸦，甚至用运动和线条涂鸦。也许他们会在这样做的过程中学会更动态地思考。
The next phase of the project will produce the most spectacular effects as the birds go into motion. But the printed page cannot capture either the product or the process: the serendipitous discoveries, the bugs, and the mathematical insights all require movement to be appreciated. Reflecting on what you are missing leads me to another description of something new the computer offers a child: the opportunity to draw in motion, indeed to doodle and even to scribble with movement as well as with lines. Perhaps they will be learning, as they do so, to think more dynamically.
第四章
CHAPTER 4
计算机语言和人类语言
Languages for Computers and for People
蜈蚣很高兴
The Centipede was happy quite
直到蟾蜍在玩乐
Until the toad in fun
说，祈祷哪条腿在哪条腿后面？
Said, Pray which leg comes after which?
这让她心烦意乱
This wrought her mind to such a pitch
她心烦意乱地躺在沟里
She lay distracted in a ditch
考虑如何运行
Considering how to run
-匿名的
—ANONYMOUS
蜈蚣的故事令人不安。我们通常认为，思考和理解从定义上来说都是好事，尤其对学习大有裨益。但蜈蚣却因思考自己的行为而遭遇不幸。同样的事情会发生在我们身上吗？这是否意味着我们应该放弃思考自己？事实上，在我们的“理性”文化中，思考阻碍行动，甚至思考阻碍学习的观念相当普遍。我们通常这样谈论学骑自行车：“继续尝试——总有一天你会‘学会’的。”这是父母对那些努力骑两轮车的孩子的标准建议。
THE CENTIPEDE STORY IS DISTURBING. WE USUALLY LIKE TO THINK that thinking and understanding are, by definition, good things to do, and that, in particular, they are useful in learning. But the centipede came to grief by thinking about her own actions. Would the same thing happen to us? Does this mean we should give up thinking about ourselves? In fact, in our “rational” culture, the notion that thinking impedes action, even that thinking impedes learning, is quite prevalent. It is our usual way of talking about learning to ride a bicycle: “Keep trying—one day you’ll just ‘get it’” is standard parental advice to children struggling with the two-wheeler.
许多哲学家提出了这样一种观点，即有些知识无法用语言描述或通过有意识的思维来掌握。这种观点被积极学习的倡导者带入了最近的课程改革，并得到了 JS Bruner 的第 1 种有影响力的认知方式分类的理论支持：有些知识以行动来表示，有些以图像来表示，只有第三类知识才以符号来表示。布鲁纳断言，“文字和图表”对于表示某些只能以行动来表示的知识是“无能为力的”。在本章中，我试图对这些问题提出更灵活的看法。
Many philosophers have developed the idea that some knowledge cannot be described in words or grasped by conscious thought. The idea was brought into recent curriculum reforms by advocates of active learning and given theoretical support by J. S. Bruner’s1 influential classification of ways of knowing: Some knowledge is represented as action, some as image, and only the third category as symbols. Bruner has asserted that “words and diagrams” are “impotent” to represent certain kinds of knowledge which are only representable as action. In this chapter I try to develop a more flexible perspective on these problems.
我的观点更加灵活，因为它拒绝了可言说与不可言说的二分法。没有一种知识可以完全用语言来表达，也没有一种知识是完全无法言说的。我的观点也更加灵活，因为它认识到了一个历史维度：知识史的一个重要组成部分是发展出一些能够提高“文字和图表”效力的技术。历史上的事实对个人来说也是如此：成为一个好的学习者的一个重要部分是学习如何拓展我们能用语言表达的范围。从这个角度来看，关于自行车的问题不在于我们能否“完整地”告诉别人如何骑车，而在于我们可以做些什么来提高我们与他人沟通的能力（以及与自己进行内心对话的能力），以便对学习骑车产生影响。本章的中心主题是发展用于谈论学习的描述性语言。我们将特别关注许多人认为最好通过“直接去做”来完成的一种学习——学习身体技能。我们的做法与学校对待“体育”——非智力科目——的方式完全相反。我们的策略是让孩子们也能明白，学习体育技能与建立科学理论有很多共同之处。
My perspective is more flexible because it rejects the idea of the dichotomy verbalizable versus nonverbalizable. No knowledge is entirely reducible to words, and no knowledge is entirely ineffable. My perspective is more flexible also in recognizing a historical dimension: An important component in the history of knowledge is the development of techniques that increase the potency of “words and diagrams.” What is true historically is also true for the individual: An important part of becoming a good learner is learning how to push out the frontier of what we can express with words. From this point of view the question about the bicycle is not whether or not one can “tell” someone “in full” how to ride but rather what can be done to improve our ability to communicate with others (and with ourselves in internal dialogues) just enough to make a difference to learning to ride. The central theme of this chapter is the development of descriptive languages for talking about learning. We shall focus particular attention on one of the kinds of learning that many people believe to be best done by “just doing it”—the learning of physical skills. Our approach to this is the exact opposite of the way schools treat “physical education”—as a nonintellectual subject. Our strategy is to make visible even to children the fact that learning a physical skill has much in common with building a scientific theory.
认识到这一点会带来很多好处。首先，我从LOGO实验室的工作中了解到，这意味着更有效地学习身体技能。2如果没有这种直接的好处，试图通过与身体活动进行类比来“激发”科学思想很容易沦为“教师模棱两可的谈话”的另一个例子。但是，如果我们能在孩子认为重要且个人的活动中找到一个科学思维的诚实位置，我们将打开更连贯、更协调的学习模式之门。
With this realization comes many benefits. First, I know from work in the LOGO laboratory that it means more effective learning of physical skills.2 Without this direct benefit, seeking to “motivate” a scientific idea by drawing an analogy with a physical activity could easily degenerate into another example of “teacher’s double talk.” But if we can find an honest place for scientific thinking in activities that the child feels are important and personal, we shall open the doors to a more coherent, syntonic pattern of learning.
在本章中，我将说明这是可以做到的，并提出将科学与身体技能联系起来对学习科学的帮助远不止提供教育者所谓的“动机”。它可以让孩子们通过了解科学家使用正式的描述性语言，并了解他们也可以使用这些语言作为学习身体技能（例如杂耍）的工具，从而对科学家产生某种认同感。这个想法是让孩子们在用身体做一些令人愉快的事情时，认为自己是在“做科学”。如果孩子们能把笛卡尔的坐标几何发明看作与他们自己的日常生活经历并非完全陌生的东西，这不仅可以让笛卡尔更有意义，而且可以帮助孩子们认识到自己更有意义。
In this chapter I show that this can be done and suggest that relating science to physical skills can do much more for learning science than providing what educators like to call a “motivation.” It can potentially place children in a position of feeling some identification with scientists through knowing that scientists use formal descriptive languages and knowing that they too can use such languages as tools for learning physical skills—juggling for example. The idea is to give children a way of thinking of themselves as “doing science” when they are doing something pleasurable with their bodies. If children could see Descartes’s invention of coordinate geometry as something not totally alien to their own experiences of daily life, this could not only make Descartes more meaningful but, at the same time, help the children come to see themselves as more meaningful.
让我们更仔细地看看我们的文化对学习身体技能的看法。在这一点上，它并不比我们之前讨论过的更“抽象”学科的数学更一致。尽管大众的观点和大部分教育心理学可能都认为学习身体技能是一个“有意识的”思考无济于事的领域，但以体育运动为生的人们并不总是同意这一点。一些最成功的教练投入了巨大的努力来分析和表达必须学习和完善的动作。体育作家蒂莫西·加尔韦将大众对这一矛盾的敏感转化为出版成功。在他的书《内心的网球》中，他提出了一些摆脱困境的建议。加尔韦鼓励学习者将自己视为由两个自我组成：一个分析性的、语言的自我，以及一个更全面、直觉的自我。他认为，合适的做法是，现在这两个自我中的一个和另一个应该掌控一切；事实上，学习过程的一个重要部分是教会每个“自我”知道何时接管以及何时将其交给另一个自我。
Let us look a bit more closely at what our culture thinks about learning physical skills. It is no more consistent regarding this than it is regarding the mathematics of more “abstract” subjects we discussed earlier. Although the popular wisdom and much of educational psychology may agree that learning physical skills is a domain where “conscious” thinking doesn’t help, people who make sports their livelihood don’t always agree. Some of the most successful coaches put great effort into analyzing and verbalizing the movements that must be learned and perfected. One sportswriter, Timothy Gallwey, has turned popular sensitivity to this contradiction into publishing success. In his book Inner Tennis he offers some suggestions for a way out of the dilemma. Gallwey encourages the learner to think of himself as made up of two selves: an analytic, verbal self and a more holistic, intuitive one. It is appropriate, he argues, that now one and now the other of these two selves should be in control; in fact, an important part of the learning process is teaching each “self” to know when to take over and when to leave it to the other.
加尔韦对成功学习过程中的协商和交易的描述在教育界中并不常见。在分析和整体思维模式的选择上，他把控制权交给了学习者。这与学校课程设计中通常发生的情况大不相同。课程改革者经常关注口头和非口头实验学习之间的选择。但他们的策略通常是从上而下做出选择，并将其纳入课程。加尔韦的策略是帮助学习者学会如何自己做出选择，这一观点符合已经提出的儿童作为认识论者的观点，鼓励儿童成为识别和选择各种思维方式的专家。
Gallwey’s description of the negotiation and transactions that go with successful learning is unusual in educational circles. In the choice between analytic and holistic modes of thinking, he gives control to the learner. This is very different from what usually happens in curriculum design for schools. Curriculum reformers are often concerned about the choice between verbal and nonverbal experimental learning. But their strategy is usually to make the choice from above and build it into the curriculum. Gallwey’s strategy is to help learners learn how to make the choice for themselves, a perspective that is in line with the vision already suggested of the child as epistemologist, where the child is encouraged to become expert in recognizing and choosing among varying styles of thought.
以蒂莫西·高威为例，并不意味着赞同他所说的一切。他的大多数想法在我看来都是有问题的。但我认为他非常正确地认识到，人们需要更结构化的方式来谈论和思考技能的学习。当代语言在这方面还不够丰富。
Taking Timothy Gallwey as an example is not an endorsement of everything he says. Most of his ideas strike me as problematic. But I think he is quite right in recognizing that people need more structured ways to talk and think about the learning of skills. Contemporary language is not sufficiently rich in this domain.
在一个计算机丰富的世界里，计算机语言既提供了控制计算机的手段，又提供了新的、强大的思维描述语言，无疑将被带入大众文化。它们将对我们描述自己和学习的语言产生特殊的影响。在某种程度上，这种情况已经发生了。对计算机一无所知的人使用“输入”、“输出”和“反馈”等概念来描述自己的心理过程并不罕见。我们将通过展示如何将编程概念用作学习特定身体技能（即杂耍）的概念框架来举例说明这一过程。因此，我们将编程视为描述手段的来源，也就是说，将其视为强化语言的一种手段。
In a computer-rich world, computer languages that simultaneously provide a means of control over the computer and offer new and powerful descriptive languages for thinking will undoubtedly be carried into the general culture. They will have a particular effect on our language for describing ourselves and our learning. To some extent this has already occurred. It is not uncommon for people with no knowledge of computers to use such concepts as “input,” “output,” and “feedback” to describe their own mental processes. We shall give an example of this process by showing how programming concepts can be used as a conceptual framework for learning a particular physical skill, namely, juggling. Thus we look at programming as a source of descriptive devices, that is to say as a means of strengthening language.
许多科学和数学进步都发挥了类似的语言功能，为我们提供了词语和概念来描述以前似乎过于模糊而无法系统思考的事物。描述性语言力量最引人注目的例子之一是解析几何的诞生，它在现代科学的发展中发挥了决定性的作用。
Many scientific and mathematical advances have served a similar linguistic function by giving us words and concepts to describe what had previously seemed too amorphous for systematic thought. One of the most striking examples of the power of descriptive language is the genesis of analytic geometry, which played so decisive a role in the development of modern science.
传说笛卡尔在一天深夜躺在床上观察天花板上的一只苍蝇时发明了解析几何。我们可以想象他当时的想法。这只苍蝇飞来飞去，其轨迹与欧几里得数学中的圆和椭圆一样真实，但无法用欧几里得语言描述。然后笛卡尔找到了一种理解它的方法：每一刻苍蝇的位置都可以用它与墙壁的距离来描述。空间中的点可以用数字对来描述；路径可以用一个方程或关系来描述，这些方程或关系对路径上的点的数字对都成立。当笛卡尔意识到如何使用代数语言来谈论空间，如何使用空间语言来谈论代数现象时，符号的效力就向前迈进了一步。笛卡尔从这一洞察中产生的坐标几何方法提供了工具，科学从此用它来描述苍蝇和行星的路径以及更抽象的物体（纯数学的东西）的“路径”。
Legend has it that Descartes invented analytic geometry while lying in bed late one morning observing a fly on the ceiling. We can imagine what his thinking might have been. The fly moving hither and thither traced a path as real as the circles and ellipses of Euclidean mathematics, but one that defied description in Euclidean language. Descartes then saw a way to grasp it: At each moment the fly’s position could be described by saying how far it was from the walls. Points in space could be described by pairs of numbers; a path could be described by an equation or relationship that holds true for those number pairs whose points lie on the path. The potency of symbols took a leap forward when Descartes realized how to use an algebraic language to talk about space, and a spatial language to talk about algebraic phenomena. Descartes’s method of coordinate geometry born from this insight provided tools that science has since used to describe the paths of flies and planets and the “paths” of the more abstract objects, the stuff of pure mathematics.
笛卡尔的突破与“乌龟圈”事件中孩子的经历有很多相似之处。我们记得，那个孩子当时明确地在寻找一种描述绕圈行走过程的方法。在LOGO中，这种描述采用了非常简单的形式，孩子只需发明描述。笛卡尔必须做更多。但在这两种情况下，回报都是能够分析性地描述此前以整体、感知-运动方式已知的事物。孩子和笛卡尔都没有遭遇蜈蚣的命运：在知道如何像以前一样分析性地描述他们的运动后，他们也能绕圈行走。
Descartes’s breakthrough has much in common with the experience of the child in the Turtle circle episode. The child, we recall, was explicitly looking for a way to describe the process of walking in a circle. In LOGO this description takes a very simple form, and the child has to invent only the description. Descartes had to do more. But in both cases the reward is the ability to describe analytically something that until then was known in a global, perceptual-kinesthetic way. Neither the child nor Descartes suffered the fate of the centipede: Both could walk in circles as well after knowing how to describe their movements analytically as before.
但是，笛卡尔的形式主义，虽然对于描述物理世界中的过程非常强大，但却不是描述物理技能世界中的过程所需要的。
But Descartes’s formalism, powerful as it is for describing processes in the world of physics, is not what is needed for describing processes in the world of physical skills.
使用微积分来描述杂耍或蜈蚣如何行走确实会让人困惑。尝试在学习身体技能时使用此类描述很可能会让学习者在最近的沟渠中焦躁不安。这种形式描述模式不适合这项任务。但其他形式主义却适合。
Using calculus to describe juggling or how a centipede walks would indeed be confusing. Attempts to use such descriptions in learning physical skills very likely would leave the learner lying with feverish mind in the nearest ditch. This mode of formal description is not matched to this task. But other formalisms are.
教育研究领域尚未朝着开发此类形式主义的方向努力。但另一个研究团体，即计算机科学家，却（出于自身原因）不得不研究描述性语言问题，并因此成为教育创新的意外资源。计算机被要求做很多事情，而要让计算机做某事，需要在某种程度上以足够的精度描述底层过程，以便机器能够执行。因此，计算机科学家投入了大量的才华和精力来开发强大的描述性形式主义。人们甚至可以说，计算机科学的称呼是错误的：它的大部分不是计算机科学，而是描述和描述性语言的科学。计算机科学产生的一些描述性形式主义正是掌握学习身体技能的过程所需要的。在这里，我们通过选择编程中的一组重要思想来证明这一点：结构化编程的概念，我们将通过五年级学生在LOGO环境中的学习体验来说明这一点。
The field of education research has not worked in the direction of developing such formalisms. But another research community, that of computer scientists, has had (for its own reasons) to work on the problem of descriptive languages and has thereby become an unexpected resource for educational innovation. Computers are called upon to do many things, and getting a computer to do something requires that the underlying process be described, on some level, with enough precision to be carried out by the machine. Thus computer scientists have devoted much of their talent and energy to developing powerful descriptive formalisms. One might even say that computer science is wrongly so called: Most of it is not the science of computers, but the science of descriptions and descriptive languages. Some of the descriptive formalisms produced by computer science are exactly what are needed to get a handle on the process of learning a physical skill. Here we demonstrate the point by choosing one important set of ideas from programming: the concept of structured programming, which we shall illustrate by the learning experience of a fifth grader in a LOGO environment.
基思给自己设定的目标是让计算机画出一个如方框“目标”所示的火柴人（见图10a）。
Keith had set himself the goal of making the computer draw a stick figure as in the box GOAL (see Figure 10a).
他的计划是从一只脚开始，画出方框序列中所示的乌龟笔画。在这样做时，他使用了一个他在前计算文化中熟悉的图像，在那里他学会了连点画，并学会了逐步描述他的活动。所以他在这里采用这种方法是完全自然的。这项任务看起来很简单，但有点乏味。他写道（图 10b）：
His plan was to start with one foot and draw the Turtle strokes illustrated in the box SEQUENCE. In doing so he is using an image familiar in his precomputational culture, where he has learned to do connect-the-dots drawing and to describe his activities in a step-by-step way. So it is perfectly natural for him to adopt this method here. The task seemed simple if somewhat tedious. He wrote (Figure 10b):
图 10a。目标
Figure 10a. Goal
图 10b
Figure 10b
图 10c. 被窃听的人
Figure 10c. Bugged Man
屏幕上出现的是一幅完全出乎意料的“BUGGED MAN”图画（见图 10c）。到底出了什么问题？
What appeared on the screen was the totally unexpected drawing of the BUGGED MAN (see Figure 10c). What went wrong?
基思对这种意外情况做好了准备。如前所述，LOGO环境的主要支柱之一是与“错误”和“调试”相关的概念集群。人们不会指望第一次尝试就能成功。人们不会以“正确——你会得到好成绩”和“错误——你会得到坏成绩”这样的标准来判断。相反，人们会问这样一个问题：“我该如何修复它？”要修复它，人们首先必须了解发生了什么。只有这样，我们才能按照我们的方式让它发生。但在这种情况下，基思无法弄清楚发生了什么。他编写程序的方式使得很难找出他的错误。他的程序中的错误在哪里？什么错误会导致他想要的结果发生如此巨大的变化？
Keith was prepared for surprises of this sort. As mentioned earlier, one of the mainstays of the LOGO environment is the cluster of concepts related to “bugs” and “debugging.” One does not expect anything to work at the first try. One does not judge by standards like “right—you get a good grade” and “wrong—you get a bad grade.” Rather one asks the question: “How can I fix it?” and to fix it one has first to understand what happened in its own terms. Only then can we make it happen on our terms. But in this situation Keith was unable to figure out what had happened. He had written his program in such a way that it was extremely difficult to pinpoint his error. Where was the bug in his program? What error could cause such a wild transformation of what he had intended?
为了理解他的困境，我们将他的程序与另一种称为“结构化编程”的编程策略进行了对比。我们的目标是将程序细分为自然的部分，以便我们可以分别调试每个部分的程序。在 Keith 的冗长而毫无特色的指令集中，很难发现和捕获错误。但是，通过使用小部分，可以限制错误并更容易捕获和找出错误。在这种情况下，自然的细分是编写一个程序来绘制V形实体以用作手臂和腿，另一个程序绘制一个正方形作为头部。一旦编写并测试了这些“子程序”，编写“超级程序”来绘制火柴人本身就非常容易了。我们可以编写一个非常简单的程序来绘制火柴人：
In order to understand his predicament we contrast his program with a different strategy of programming known as “structured programming.” Our aim is to subdivide the program into natural parts so that we can debug programs for each part separately. In Keith’s long, featureless set of instructions it is hard to see and trap a bug. By working with small parts, however, bugs can be confined and more easily trapped, figured out. In this case a natural subdivision is to make a program to draw a V-shaped entity to use for arms and legs and another to draw a square for the head. Once these “subprocedures” have been written and tested, it is extremely easy to write the “superprocedure” to draw the stick figure itself. We can write an extremely simple program to draw the stick figure:
对男人
TO MAN
维珍妮
VEE
前进 50
FORWARD 50
维珍妮
VEE
前锋 25
FORWARD 25
头
HEAD
结尾
END
这个过程很简单，很容易理解。但当然，只有假设计算机能够理解命令VEE和HEAD ，它才能实现其简单性。如果计算机不能理解命令，那么下一步必须定义VEE和HEAD。我们可以用相同的方式做到这一点，即始终使用我们可以整体理解的过程。例如：
This procedure is simple enough to grasp as a whole. But of course it achieves its simplicity only by making the assumption that the commands VEE and HEAD are understood by the computer. If they are not, the next step must be to define VEE and HEAD. We can do this in the same style of always working with a procedure we can understand as a whole. For example:
致 VEE
TO VEE
右 120
RIGHT 120
LINE 50
LINE 50
右 120
RIGHT 120
LINE 50
LINE 50
右 120
RIGHT 120
结尾
END
（在这个程序中，我们假设我们已经定义了命令LINE，它使 Turtle 前进和后退。）
(In this program we assume we have defined the command LINE, which causes the Turtle to go forward and come back.)
为了实现这一点，我们接下来定义LINE：
To make this work we next define LINE:
至 线:距离
TO LINE:DISTANCE
前锋：距离
FORWARD:DISTANCE
后卫：距离
BACK:DISTANCE
结尾
END
由于最后一个过程仅使用固有的LOGO命令，因此无需进一步定义即可工作。为了完成MAN，我们通过以下方式定义HEAD：
Since the last procedure uses only innate LOGO commands, it will work without further definitions. To complete MAN we define HEAD by:
前往
TO HEAD
右 45
RIGHT 45
20 平方
SQUARE 20
结尾
END
七年级学生罗伯特 (Robert) 表达了他对这种编程风格的转变，他惊呼道：“看，我所有的程序都是思维大小的碎片。”罗伯特通过以下评论进一步阐述了这个比喻：“我过去常常被我的程序搞得一头雾水。现在我不会贪多嚼不烂。”他遇到了一个强大的想法：有可能建立一个庞大的智力系统，而不必迈出无法理解的一步。使用分层结构进行构建使人们能够掌握整个系统，也就是说，可以“从顶部看”系统。
Robert, a seventh grader, expressed his conversion to this style of programming by exclaiming: “See, all my procedures are mind-sized bites.” Robert amplified the metaphor by comments such as: “I used to get mixed up by my programs. Now I don’t bite off more than I can chew.” He had met a powerful idea: It is possible to build a large intellectual system without ever making a step that cannot be comprehended. And building with a hierarchical structure makes it possible to grasp the system as a whole, that is to say, to see the system as “viewed from the top.”
非结构化MAN程序的作者 Keith曾接触过使用子程序的想法，但之前一直拒绝使用。“直线”形式的程序更符合他熟悉的做事方式。直到有一天他无法调试他的MAN程序，他才感受到对结构化编程的迫切需要。在LOGO环境中，我们一次又一次地看到这种情况发生。当处于这种困境的孩子问该怎么做时，通常只需说：“你知道该怎么做！”而且孩子经常会说，有时是得意洋洋，有时是羞怯地：“我想我应该把它变成子程序？”“正确的方法”并没有强加给 Keith；计算机给了他足够的灵活性和权力，使他的探索可以是真实的，也是他自己的。
Keith, the author of the nonstructured MAN program, had been exposed to the idea of using subprocedures but had previously resisted it. The “straight-line” form of program corresponded more closely to his familiar ways of doing things. He had experienced no compelling need for structured programming until the day he could not debug his MAN program. In LOGO environments we have seen this happen time and again. When a child in this predicament asks what to do, it is usually sufficient to say: “You know what to do!” And often the child will say, sometimes triumphantly, sometimes sheepishly: “I guess I should turn it into subprocedures?” The “right way” was not imposed on Keith; the computer gave him enough flexibility and power so that his exploration could be genuine and his own.
这两种项目规划和实施方式非常普遍。通过观察学习“身体”技能和“智力”技能的方式，我们可以看出这两种方式。例如，我们来看看两名五年级学生的情况，他们在我们的儿童学习实验室里学习了编程和身体技能。
These two styles of approaching the planning and working out of a project are pervasive. They can be seen by observing styles of learning “physical” as well as “intellectual” skills. Consider, for example, the case of two fifth graders who learned both programming and physical skills in our children’s learning laboratory.
迈克尔体格健壮，运动能力强，在他自己眼中是个“硬汉”。保罗则比较内向，好学，体型瘦小。迈克尔在学校的表现很差，而保罗则表现很好，所以当保罗在电脑上进步很快，迅速进入相当复杂的结构化编程程序时，两人都没有感到惊讶。几周后，迈克尔仍然只能以直线式风格编写程序。毫无疑问，他拥有编写更复杂程序所需的所有概念，但他对使用子程序有着强烈的抵触情绪。
Michael is strong, athletic, a “tough kid” in his own eyes. Paul is more introverted, studious, slightly built. Michael does poorly at school and Paul does well, so when Paul got on faster in work with the computer, moving quickly into quite complex structured programming procedures, neither one was surprised. After several weeks Michael was still able to write programs only in the straight-line style. There was no doubt that he possessed all the necessary concepts to write more elaborate programs, but he was held back by a classical and powerful resistance to using subprocedures.
这时，两人开始练习踩高跷。迈克尔的策略是在脑海中固定一个按顺序踩高跷的模型：“脚踩在横杆上，站起身，脚踩在另一根横杆上，第一只脚向前……”当尝试这样做导致快速摔倒时，他会勇敢地一次又一次地重新开始，相信他最终会成功，事实上他确实成功了。但令他们两人都感到惊讶的是，保罗先到了那里。
At this time both began to work on stilt walking. Michael’s strategy was to fix in his mind a model of stilt walking in sequential form: “Foot on the bar, raise yourself up, foot on the other bar, first foot forward…” When attempting to do it led to a rapid crash, he would bravely start again and again and again, confident that he would eventually succeed, which in fact he did. But, to the surprise of both of them, Paul got there first.
保罗的策略有所不同。他以同样的方式开始，但当他发现没有取得进展时，他试图隔离并纠正造成问题的部分过程：“错误”。当你向前迈进时，你往往会把高跷抛在身后。一旦发现这个错误，根除它并不难。这样做的一个技巧是想象用高跷而不是用脚迈出一步，让高跷“携带”脚。这是通过用手臂抵住脚来抬起高跷来实现的。保罗认为这与他编程方法的类比是如此明显，这可能是从编程工作“转移”到学习这种身体技能的一个例子。
Paul’s strategy was different. He began in the same way but when he found that he was not making progress he tried to isolate and correct part of the process that was causing trouble: “the bug.” When you step forward you tend to leave the stilt behind. This bug, once identified, is not hard to eradicate. One trick for doing so is to think of taking the step with the stilt rather than with the foot and let the stilt “carry” the foot. This is done by lifting the stilt with the arm against the foot. The analogy with his approach to programming was so apparent to Paul that this might have been a case of “transfer” from the programming work to learning this physical skill.
实际上，这两种情况更有可能都借鉴了他长期以来的认知风格。但LOGO的经验确实使保罗能够清晰地表达他的风格的这些方面。在迈克尔的案例中，编程和踩高跷之间的关系更加清晰。只有通过这种类比，他才明确认识到他和保罗踩高跷风格之间的差异！换句话说，编程的经验帮助两个男孩更好地掌握了自己的行为，对自己有了更清晰的认识。
Actually, it is more likely that both situations drew on long-standing features of his general cognitive style. But the experience with LOGO did enable Paul to articulate these aspects of his style. The relation between programming and stilt walking was even clearer in Michael’s case. It was only through this analogy that he came to recognize explicitly the difference between his and Paul’s style of going about the stilt walking! In other words the experience of programming helped both boys obtain a better grasp of their own actions, a more articulated sense of themselves.
结构化编程作为一种数学原理，即学习辅助手段，其普遍性将在下一个例子中变得更加明显，这个例子描述了学习另一项身体技能——杂耍的过程。我们并不是随机选择的。海龟圈是“用身体”学习数学的良好载体。事实证明，杂耍也是“用数学”学习身体技能的同样好的载体。当然，情况更加复杂，也更加有趣，因为在这两种情况下，这个过程都是双向的，从计算隐喻到肢体语言，再回到计算隐喻。在体验海龟几何的过程中，孩子们对自己的身体和身体动作的感觉以及对形式几何的理解都更加敏锐。而关于结构化程序的理论思想，当用杂耍术语——真实的身体术语——来表达时，就会呈现出新的具体性和力量。在这两种情况下，知识都具有我们称之为共振的性质。
The generality of the idea of structured programming as a mathetic principle, that is to say an aid to learning, will become more apparent through the next example, which describes the process involved in learning another physical skill—juggling. We do not choose it at random. The Turtle circle was a good carrier for learning mathematics “with one’s body.” Juggling turns out to be an equally good carrier for learning a body skill “with mathematics.” Of course, the picture is more complicated and also more interesting because in both cases the process works in both directions, from computational metaphor to body language and back again. In passing through an experience of Turtle geometry, children sharpen their sense of their bodies and their physical movements as well as their understanding of formal geometry. And theoretical ideas about structured programs, when couched in juggling terms—real body terms—take on new concreteness and power. In both cases, knowledge takes on the quality we have characterized as syntonic.
杂耍有很多种。大多数人想到杂耍时，都会想到一种叫做“淋浴杂耍”的程序。在淋浴杂耍中，球在“圆圈”中一个接一个地移动，从顶部的左到右，从底部的右到左（或反之亦然）。这需要两种抛掷方式：一种是短而低的抛掷，将球从一只手抛到“圆圈”底部（靠近手）的另一只手；另一种是长而高的抛掷，使球绕着圆圈的顶部旋转。（见图11。）
There are many different kinds of juggling. When most people think of juggling, they are thinking about a procedure that is called “showers juggling.” In showers juggling, balls move one behind the other in a “circle” passing from left to right at the top and from right to left at the bottom (or vice versa). This takes two kinds of throws: a short, low throw to get the balls from one hand to the other at the bottom of the “circle” (near the hands), and a long, high throw to get the balls to go around the top of the circle. (See Figure 11.)
级联杂耍的结构更简单。没有圆圈的底部；球在弧形上方以两个方向移动。只有一种抛掷方式：长抛高抛。（见图11。）它的简单性使它成为杂耍的更好途径，也是我们论证的更好例子。我们的指导问题是：对于希望学习级联杂耍的人，口头、分析性的描述会对其有所帮助还是会阻碍其学习？答案是：这完全取决于学习者用于进行分析性描述的材料。我们使用级联杂耍来展示良好的计算模型如何帮助构建提高技能表现的“人员程序”，以及对这些人员程序的反思如何帮助我们学习编程和数学。但是，当然，一些口头描述会造成更多的困惑而不是帮助。例如，考虑以下描述：
Cascade juggling has a simpler structure. There is no bottom of the circle; balls travel in both directions over the upper arc. There is only one kind of toss: a long and high one. (See Figure 11.) Its simplicity makes it a better route into juggling as well as a better example for our argument. Our guiding question is this: Will someone who wishes to learn cascade juggling be helped or hindered by a verbal, analytic description of how to do it? The answer is: It all depends. It depends on what materials the learner has for making analytic descriptions. We use cascade juggling to show how good computational models can help construct “people procedures” that improve performance of skills and how reflection on those people procedures can help us learn to program and to do mathematics. But, of course, some verbal descriptions will confuse more than they will help. Consider, for example, the description:
图 11. 两种形式的杂耍
Figure 11. Two Forms of Juggling
1. 先用左手拿 1 号球和 2 号球，用右手拿 3 号球。
1. Start with balls 1 and 2 in the left hand and ball 3 in the right.
2. 将球 1 以高抛物线形式向右手抛出。
2. Throw ball 1 in a high parabola to the right hand.
3. 当 1 号球到达顶点时，将 3 号球以类似的高抛物线向左手抛出，但要注意将 3 号球抛到 1 号球的轨迹下方。
3. When ball 1 is at the vertex throw ball 3 over to the left hand in a similar high parabola, but take care to toss ball 3 under the trajectory of ball 1.
4. 当 1 号球到达右手边，3 号球在顶点时，接住 1 号球，并按照 3 号球轨迹下方的轨迹抛出 2 号球，依此类推。
4. When ball 1 arrives at the right hand and ball 3 is at the vertex, catch ball 1 and throw ball 2 in a trajectory under that of ball 3, and so on.
这种描述基本上是一种强力的直线程序。对于学习来说，这种描述毫无用处。计算机文化之外的人可能会说它太像计算机程序了，“只是一个接一个的指令”。它就像某些程序，例如 Keith 的第一个MAN程序。但我们已经看到，在没有良好内部结构的情况下将指令串在一起也不是计算机编程的好模型，我们将看到，结构化编程技术不仅适合编写程序，而且也适合作为杂耍的数学描述。
This description is basically a brute-force straight-line program. It is not a useful description for the purpose of learning. People outside the computer culture might say it is too much like a computer program, “just one instruction after another.” It is like certain programs, for example Keith’s first MAN program. But we have seen that stringing instructions together without good internal structure is not a good model for computer programming either, and we shall see that the techniques of structured programming that are good for writing programs are also good as mathetic descriptions of juggling.
我们的目标是创建一个人类程序：TO JUGGLE。作为定义该程序的第一步，我们识别并命名了与 Keith 绘制火柴人时使用的子程序（TO VEE、TO HEAD、TO LINE）作用类似的子程序。在玩杂耍的情况下，一对自然的子程序就是我们所说的TOSSRIGHT和TOSSLEFT。正如命令VEE 的功能定义是使计算机在屏幕上放置某个 V 形图形一样，给我们的杂耍学徒下达的命令TOSSRIGHT应该“使”他将一个球（我们假设他左手拿着球）扔到右手。
Our goal is to create a people procedure: TO JUGGLE. As a first step toward defining this procedure, we identify and name subprocedures analogous in their role to the subprocedures Keith used in drawing his stick figure (TO VEE, TO HEAD, TO LINE). In the case of juggling, a natural pair of subprocedures is what we call TOSSRIGHT and TOSSLEFT. Just as the command VEE was defined functionally by the fact that it causes the computer to place a certain V-shaped figure on the screen, the command TOSSRIGHT given to our apprentice juggler should “cause” him to throw a ball, which we assume he is holding in his left hand, over to the right hand.
但是TO MAN 的编程和TO JUGGLE的编程之间存在一个重要的区别。TO MAN的 程序员不必担心时间问题，但在设置杂耍程序时我们必须担心时间问题。杂耍者必须在一个周期的适当时刻执行TOSSRIGHT和TOSSLEFT动作，并且这两个动作必须在时间上重叠。由于我们选择将接球阶段与投掷阶段包含在同一个子程序中，因此程序TOSSRIGHT意味着包括在球飞到右手时接球。同样，TOSSLEFT是一个命令，要求将球从右手扔到左手并在球到达时接住它。3
But there is an important difference between programming TO MAN and programming TO JUGGLE. The programmer of TO MAN need not worry about timing, but in setting up the procedure for juggling we must worry about it. The juggler must perform the actions TOSSRIGHT and TOSSLEFT at appropriate moments in a cycle, and the two actions will have to overlap in time. Since we have chosen to include the catching phase in the same subprocedure as the throwing phase, the procedure TOSSRIGHT is meant to include catching the ball when it comes over to the right hand. Similarly, TOSSLEFT is a command to throw a ball from the right hand over to the left and catch it when it arrives.3
由于大多数人都能执行这些操作，我们将TOSSLEFT和TOSSRIGHT视为既定的，并集中精力研究如何将它们组合起来形成一个新过程，我们称之为TO JUGGLE 。将它们组合在一起与将子过程TO VEE和TO HEAD组合起来形成过程TO MAN有一个本质区别。在前一个TOSSRIGHT发起的操作完成之前，可能必须先发起TOSSLEFT。用计算机科学的语言来表达，这可以表示为，我们正在处理并行过程，而不是绘制火柴人时使用的严格串行过程。
Since most people can perform these actions, we shall take TOSSLEFT and TOSSRIGHT as given and concentrate on how they can be combined to form a new procedure we shall call TO JUGGLE. Putting them together is different in one essential way from the combination of subprocedures TO VEE and TO HEAD to make the procedure TO MAN. TOSSLEFT might have to be initiated before the action initiated by the previous TOSSRIGHT is completed. In the language of computer science, this is expressed by saying that we are dealing with parallel processes as opposed to the strictly serial processes used in drawing the stick figure.
为了描述子程序的组合，我们引入了一个新的编程元素：“WHEN DEMON”的概念。这可以通过以下指令来说明：WHEN HUNGRY EAT 。在LOGO的一个版本中，这意味着：每当发生称为HUNGRY的条件时，执行称为EAT的操作。“恶魔”的隐喻表达了这样一种观点：该命令在计算机系统内创建了一个自主实体，该实体保持休眠状态，直到发生某种事件，然后像恶魔一样，它会突然出现并执行其操作。杂耍表演将使用两个这样的WHEN DEMONS。
To describe the combination of the subprocedures we introduce a new element of programming: the concept of a “WHEN DEMON.” This is illustrated by the instruction: WHEN HUNGRY EAT. In one version of LOGO this would mean: Whenever the condition called HUNGRY happens, carry out the action called EAT. The metaphor of a “demon” expresses the idea that the command creates an autonomous entity within the computer system, one that remains dormant until a certain kind of event happens, and then, like a demon, it pounces out to perform its action. The juggling act will use two such WHEN DEMONS.
它们的定义如下：
Their definitions will be something like:
当某物被抛出
WHEN something TOSSLEFT
当某事发生时
WHEN something TOSSRIGHT
为了填补空白，即“某些事情”，我们描述了两种条件，或可识别的系统状态，它们将触发抛掷动作。
To fill the blanks, the “somethings,” we describe two conditions, or recognizable states of the system, that will trigger the tossing action.
在循环的关键时刻，球的排列方式如下（图 12）：
At a key moment in the cycle the balls are disposed about like this (Figure 12):
图 12
Figure 12
但此系统状态图并不完整，因为它未能显示顶部球的飞行方向。为了完善它，我们添加箭头来指示方向（图 13a），并获得两个状态描述（图 13b和13c）。
But this diagram of the state of the system is incomplete since it fails to show in which direction the top ball is flying. To complete it we add arrows to indicate a direction (Figure 13a) and obtain two state descriptions (Figures 13b and 13c).
图 13a
Figure 13a
图 13b。右上方：球在上方并向右移动
Figure 13b. TOPRIGHT: The ball is at the top and is moving to the right
图 13c。左上角：球在顶部，向左移动
Figure 13c. TOPLEFT: The ball is at the top and is moving to the left
如果我们合理地假设玩杂耍的人能够识别这两种情况，那么以下形式应该是不言而喻的：
If we assume, reasonably, that the juggler can recognize these two situations, the following formalism should be self-explanatory:
继续玩杂耍
TO KEEP JUGGLING
当右转时
WHEN TOPRIGHT TOSSRIGHT
当左上角向左转
WHEN TOPLEFT TOSSLEFT
或者更简单一点：
or even more simply:
继续玩杂耍
TO KEEP JUGGLING
当TOPX TOSSX
WHEN TOPX TOSSX
它声明当TOPRIGHT状态发生时，右手应该开始抛掷，当TOPLEFT 状态发生时，左手应该开始抛掷。稍加思考就会发现这是一个完整的描述：杂耍过程将以自我延续的方式继续，因为每次抛掷都会创建一个触发下一次抛掷的系统状态。
which declares that when the state TOPRIGHT occurs, the right hand should initiate a toss and when TOPLEFT occurs, the left hand should initiate a toss. A little thought will show that this is a complete description: The juggling process will continue in a self-perpetuating way since each toss creates a state of the system that triggers the next toss.
这种将杂耍变成人际交往过程的模型如何应用于教学策略？首先，请注意杂耍模型做出了几个假设：
How can this model that turned juggling into a people procedure be applied as a teaching strategy? First, note that the model of juggling made several assumptions:
1. 学习者能够进行向右转和向左转
1. that the learner can perform TOSSRIGHT and TOSSLEFT
2. 她可以识别触发状态TOPLEFT和TOPRIGHT
2. that she can recognize the trigger states TOPLEFT and TOPRIGHT
3. 她可以根据程序的定义将这些表演能力结合起来，以保持杂耍
3. that she can combine these performance abilities according to the definitions of the procedure TO KEEP JUGGLING
现在，我们将我们的假设和人员程序转化为教学策略。
Now, we translate our assumptions and our people procedure into a teaching strategy.
步骤 1.确认学习者可以抛球。给她一个球，让她把它抛到另一只手上。这通常很顺利，但我们稍后会看到，通常需要进行一些细微的改进。自发程序有一个缺陷。
STEP 1. Verify that the learner can toss. Give her one ball, ask her to toss it over into the other hand. This usually happens smoothly, but we will see later that a minor refinement is often needed. The spontaneous procedure has a bug.
步骤2. 确认学习者可以组合投掷。尝试使用两个球，并按照说明进行操作：
STEP 2. Verify that the learner can combine tosses. Try with two balls with instructions:
穿越
TO CROSS
右上方
TOPRIGHT
当右上方向左转时
WHEN TOPRIGHT TOSSLEFT
结尾
END
这是为了在左手和右手之间交换球。虽然它看起来是TOSSLEFT和TOSSRIGHT的简单组合，但它通常不会立即起作用。
This is intended to exchange the balls between left and right hands. Although it appears to be a simple combination of TOSSLEFT and TOSSRIGHT, it usually does not work immediately.
步骤3. 查找抛球程序中的错误。为什么TO CROSS不起作用？通常我们发现学习者的抛球能力并不像步骤 1 中看起来那么好。抛球程序中最常见的偏差或“错误”是在抛球时用眼睛跟着球。由于一个人只有一双眼睛，他们在第一次抛球中的投入使得第二次抛球几乎不可能，因此通常以球落地的灾难告终。
STEP 3. Look for bugs in the toss procedures. Why doesn’t TO CROSS work? Typically we find that the learner’s ability to toss is not really as good as it seemed in step 1. The most common deviation or “bug” in the toss procedure is following the ball with the eyes in doing a toss. Since a person has only one pair of eyes, their engagement in the first toss makes the second toss nearly impossible and thus usually ends in disaster with the balls on the floor.
步骤4。调试。假设错误是用眼睛跟踪第一个球，我们通过让学习者重新投掷一个球而不用眼睛跟踪来进行调试。大多数学习者发现（令他们惊讶的是）几乎不需要练习就能在投掷的同时将眼睛固定在飞球形成的抛物线的预期顶点周围。当单次投掷调试完成后，学习者再次尝试结合两次投掷。现在这种方法通常有效，尽管可能还有另一个错误需要消除。
STEP 4. Debugging. Assuming that the bug was following the first ball with the eyes, we debug by returning our learner to tossing with one ball without following it with her eyes. Most learners find (to their amazement) that very little practice is needed to be able to perform a toss while fixing the eyes around the expected apex of the parabola made by the flying ball. When the single toss is debugged, the learner again tries to combine two tosses. Most often this now works, although there may still be another bug to eliminate.
步骤5. 扩展到三个球。一旦学习者能够顺利执行我们称为CROSS的程序，我们将继续进行三个球练习。为此，首先一只手拿两个球，另一只手拿一个球（图 14）。
STEP 5. Extension to three balls. Once the learner can smoothly execute the procedure we called CROSS, we go on to three balls. To do this, begin with two balls in one hand and one in the other (Figure 14).
抛球 2 就像执行CROSS一样，忽略球 1。CROSS中的TOSSRIGHT使三个球进入准备进行KEEP JUGGLING的状态。从CROSS到KEEP JUGGLING的过渡对一些学习者来说有点困难，但这很容易克服。大多数人都可以使用此策略在不到半小时内学会玩杂耍。
Ball 2 is tossed as if executing CROSS, ignoring ball 1. The TOSSRIGHT in CROSS brings the three balls into a state that is ready for KEEP JUGGLING. The transition from CROSS to KEEP JUGGLING presents a little difficulty for some learners, but this is easily overcome. Most people can learn to juggle in less than half an hour by using this strategy.
图 14. 级联杂耍
Figure 14. Cascade Juggling
许多LOGO教师都使用过这种教学策略的变体，其中一位老师 Howard Austin 详细研究过这种策略，他将杂耍分析作为其博士论文的主题。毫无疑问，这种策略非常有效，其原因也毫不令人怀疑：使用编程概念作为描述性语言有助于调试。
Variants of this teaching strategy have been used by many LOGO teachers and studied in detail by one of them, Howard Austin, who took the analysis of juggling as the topic of his Ph.D. thesis. There is no doubt that the strategy is very effective and little doubt as to the cause: The use of programming concepts as a descriptive language facilitates debugging.
在我们对画火柴人和学习杂耍的描述中，一个中心主题是如何通过使用对复杂过程的适当描述来促进调试。在这两种情况下，描述都反映了以模块化形式表示的过程，也就是说，将其分解为自然的功能单元，并且通过将错误限制在尽可能狭窄的一组边界内，有助于捕获错误。当多个错误同时存在时，调试条件最差。如果模块足够小，以至于任何一个模块都不太可能包含多个错误，则调试过程尤其有效。
In our description of drawing a stick figure and of learning to juggle, a central theme was how debugging is facilitated by the use of an appropriate description of a complex process. In both cases the description reflected a representation of the process in modular form, that is to say broken up into natural, functional units, and catching the bug was helped by containing it within as narrow a set of boundaries as possible. The worst conditions for debugging are created when several bugs are present simultaneously. The debugging process is especially effective if the modules are small enough for it to be unlikely that any one contains more than one bug.
初学者尝试通过蛮力方法学习杂耍时所发生的情况很好地说明了多重错误所带来的困难。事实上（就像迈克尔学会踩高跷一样），他们经常会成功，通常是经过数小时的令人沮丧的尝试，才能让三个球保持在空中，而无法越过两个球。但他们需要很长时间才能学会。当霍华德·奥斯汀仔细观察学习者的行为时，他看到了我们在理性策略方法中描述的相同错误，例如，眼睛跟随错误。在多次重复的过程中，所谓的“反复试验”将形成一种有效的行为。纯粹是偶然的，眼睛在一次抛掷中会稍微移动一点。与其他动物一样，人类具有学习机制，能够捕捉到此类事件并增加它们再次发生的可能性。最终，错误被消除，受试者学会了杂耍。人们能够像迷宫中的老鼠一样学习。但这个过程缓慢而原始。通过有意识地控制学习过程、表达和分析我们的行为，我们可以学得更多、更快。
The difficulties produced by multiple bugs are well illustrated by what happens when beginners try to learn juggling by the brute-force approach. In fact (just as Michael learned to walk on stilts) they often succeed, usually after hours of frustrating attempts to keep three balls in the air without yet being able to cross two. But it takes them a long time to learn. When Howard Austin looked closely at the actions of the learner, he saw the same bugs that we described in our rational strategy approach, for example, the eye-following bug. In the course of very many repetitions, so-called “trial and error learning” will shape a behavior that works. By sheer chance, the eyes will happen to move a little less on one toss. Like other animals, human beings have learning mechanisms that are capable of picking up on such events and increasing the probability that they will happen again. Eventually, the bugs are ironed out and the subject learns to juggle. People are capable of learning like rats in mazes. But the process is slow and primitive. We can learn more, and more quickly, by taking conscious control of the learning process, articulating and analyzing our behavior.
计算程序增强学习能力这一事实并不意味着可以神奇地从学习中消除所有重复过程，或者学习杂耍所需的时间可以减少到几乎为零。捕捉和消除错误总是需要时间的。学习必要的组件技能也总是需要时间。可以消除的是浪费和低效的方法。如果学习者采用糟糕的学习策略，学习足够的杂耍技巧来让三个球继续旋转需要很多小时。如果采用好的策略，时间就会大大减少，通常只需二十或三十分钟。
The fact that computational procedures enhance learning does not mean that all repetitive processes can be magically removed from learning or that the time needed to learn juggling can be reduced to almost nothing. It always takes time to trap and eliminate bugs. It always takes time to learn necessary component skills. What can be eliminated are wasteful and inefficient methods. Learning enough juggling skill to keep three balls going takes many hours when the learner follows a poor learning strategy. When a good one is adopted the time is greatly reduced, often to as little as twenty or thirty minutes.
儿童经常会对调试产生一种“抵触情绪”，类似于我们对子程序化的抵触情绪。我在许多儿童第一次接触LOGO时都看到过这种情况。儿童计划让 Turtle 画出某个图形，比如房子或火柴人。他们很快编写并试用了一个程序。但程序不起作用。他们没有调试它，而是将其删除。有时整个项目都会被放弃。有时，儿童会一次又一次地尝试，他们有着令人钦佩的毅力，但总是从头开始，显然是想一次性“正确地”完成任务。儿童可能会失败，也可能会成功让计算机画出图画。但这个孩子还没有成功掌握调试策略。
Children often develop a “resistance” to debugging analogous to the resistance we have seen to subprocedurizing. I have seen this in many children’s first sessions in a LOGO environment. The child plans to make the Turtle draw a certain figure, such as a house or stick man. A program is quickly written and tried. It doesn’t work. Instead of being debugged, it is erased. Sometimes the whole project is abandoned. Sometimes the child tries again and again and again with admirable persistence but always starting from scratch in an apparent attempt to do the thing “correctly” in one shot. The child might fail or might succeed in making the computer draw the picture. But this child has not yet succeeded in acquiring the strategy of debugging.
我们很容易产生共鸣。学校的道德观念已经很好地影响了我们。我们认为一个有小错误的好程序，在孩子看来却是“错误的”、“不好的”、“失误”。学校教导我们，错误是坏的；人们最不想做的事情就是仔细研究它们、纠结它们或思考它们。孩子们很高兴利用计算机的能力来删除所有错误而不留下任何痕迹。调试哲学表明了一种相反的态度。错误对我们有益，因为它们会引导我们研究发生了什么，了解哪里出了问题，并通过理解来修复它。与其他任何活动相比，计算机编程的经验更能使孩子们“相信”调试。
It is easy to empathize. The ethic of school has rubbed off too well. What we see as a good program with a small bug, the child sees as “wrong,” “bad,” “a mistake.” School teaches that errors are bad; the last thing one wants to do is to pore over them, dwell on them, or think about them. The child is glad to take advantage of the computer’s ability to erase it all without any trace for anyone to see. The debugging philosophy suggests an opposite attitude. Errors benefit us because they lead us to study what happened, to understand what went wrong, and, through understanding, to fix it. Experience with computer programming leads children more effectively than any other activity to “believe in” debugging.
与LOGO环境的接触逐渐削弱了长期以来对调试和次程序化的抵制。一些观察到孩子们对自己“错误”的容忍度越来越高的人将这种态度的转变归因于LOGO教师，他们在孩子认为“错误”的程序面前实事求是、不加批判。我认为其中还有更根本的东西。在LOGO环境中，孩子们了解到老师也是一个学习者，每个人都会从错误中学习。
Contact with the LOGO environment gradually undermines long-standing resistances to debugging and subprocedurizing. Some people who observe the children’s growing tolerance for their “errors” attribute the change of attitude to the LOGO teachers who are matter-of-fact and uncritical in the presence of programs the child sees as “wrong.” I think that there is something more fundamental going on. In the LOGO environment, children learn that the teacher too is a learner, and that everyone learns from mistakes.
自九月份学期开始以来，一组十二名五年级学生每周都有几个小时的LOGO体验。十二月初，该小组决定开展一个集体项目。将对机械海龟进行编程，让它在学校走廊悬挂的巨型纸横幅上写上“圣诞快乐”。这是一个理想的项目。字母表中的字母被分配给小组成员。每个人都将为两个或三个字母、装饰性图画和整条消息编写程序，并使用字母程序作为子程序。
A group of twelve fifth graders had had several hours a week of LOGO experience since the beginning of the term in September. Early in December the group decided on a collective project. A mechanical Turtle would be programmed to write “Merry Christmas” on huge paper banners that would be strung in the school corridors. An ideal project. The letters of the alphabet were divided up among members of the group. Each would write programs for two or three letters, for decorative drawings, and for whole messages, using the letter programs as subprocedures.
但是暴风雪和其他干扰推迟了工作，当学校的最后一周到来时，横幅还没有制作完成。负责该小组的老师决定打破一般规则，自己进行一些编程。她在家里工作，没有海龟，所以第二天早上来上班时，她有一堆未调试的程序。她和孩子们一起调试它们。老师和一个孩子坐在地板上看着海龟画出一个本应是字母 R 的东西，但是倾斜的笔画放错了位置。错误在哪里？当他们一起苦思冥想时，孩子突然顿悟：“你的意思是，”他说，“你真的不知道如何修复它？”孩子还不知道该怎么说，但他得知，他和老师曾一起从事过一个研究项目。这个事件令人心酸。它讲述了这个孩子每次都参与老师们的“让我们一起做”的游戏，尽管他知道这种合作是虚构的。发现不能是陷阱；發明無法預設。
But snowstorms and other disruptions delayed the work, and when the last week of school arrived the banners had not yet been made. The instructor in charge of the group decided to break a general rule and to do some of the programming herself. She worked at home without a Turtle so when she came in the next morning she had a collection of un-debugged programs. She and the children would debug them together. The instructor and a child were on the floor watching a Turtle drawing what was meant to be a letter R, but the sloping stroke was misplaced. Where was the bug? As they puzzled together the child had a revelation: “Do you mean,” he said, “that you really don’t know how to fix it?” The child did not yet know how to say it, but what had been revealed to him was that he and the teacher had been engaged together in a research project. The incident is poignant. It speaks of all the times this child entered into teachers’ games of “let’s do that together” all the while knowing that the collaboration was a fiction. Discovery cannot be a setup; invention cannot be scheduled.
在传统的课堂上，教师确实会尝试与儿童合作，但通常材料本身不会自发产生研究问题。成人和儿童真的可以合作完成小学算术吗？使用计算机的一个非常重要的特点是教师和学习者可以进行真正的智力合作；他们可以一起尝试让计算机做这做那，并了解它实际上在做什么。教师和学习者以前从未见过的新情况经常出现，因此教师不必假装不知道。分享问题和解决问题的经验可以让孩子从成人那里学习，不是“按照老师说的做”，而是“按照老师做的做”。教师所做的事情之一就是研究问题，直到完全理解它。LOGO环境很特别，因为它提供了许多问题，小学生可以以一种在日常生活中很少见的完整性来理解这些问题。为了更充分地理解这一点，重新思考前面讨论的简单调试示例可能会有所帮助。
In traditional schoolrooms, teachers do try to work collaboratively with children, but usually the material itself does not spontaneously generate research problems. Can an adult and a child genuinely collaborate on elementary school arithmetic? A very important feature of work with computers is that the teacher and the learner can be engaged in a real intellectual collaboration; together they can try to get the computer to do this or that and understand what it actually does. New situations that neither teacher nor learner has seen before come up frequently and so the teacher does not have to pretend not to know. Sharing the problem and the experience of solving it allows a child to learn from an adult not “by doing what teacher says” but “by doing what teacher does.” And one of the things that the teacher does is pursue a problem until it is completely understood. The LOGO environment is special because it provides numerous problems that elementary schoolchildren can understand with a kind of completeness that is rare in ordinary life. To appreciate the point more fully, it may be useful to rethink the simple examples of debugging discussed earlier.
我们已经讨论了这个计划：
We have discussed the program:
到家
TO HOUSE
正方形
SQUARE
三角形
TRIANGLE
结尾
END
平方
TO SQUARE
重复4
REPEAT 4
前进100
FORWARD 100
右 90
RIGHT 90
结尾
END
三角
TO TRIANGLE
重复3
REPEAT 3
前进100
FORWARD 100
右 120
RIGHT 120
结尾
END
但是这个程序有一个错误，它把三角形画在了正方形里面而不是正方形上面。为什么？对于小孩子来说，这似乎有点不可思议。但是你可以通过遵循一条众所周知的启发式建议来弄清楚“为什么乌龟会做这种愚蠢的事情”：扮演乌龟。自己做，但假装和乌龟一样笨。找出乌龟这样做的原因几乎可以立即提出一种修复方法。例如，有人说：“乌龟转向正方形是因为三角形表示向右 转。”这种诊断本身就有一种解决方法（几种同样简单的方法之一）：做一个向左转的三角形程序。
But this program contains a bug and draws the triangle inside the square instead of on it. Why? It might seem mysterious at first to a child. But you can figure out “why the Turtle did that dumb thing” by following through on a already well-known piece of heuristic advice: Play Turtle. Do it yourself but pretend to be as dumb as the Turtle. Finding out why the Turtle did it almost immediately suggests a way to fix it. For example, some say: “The Turtle turned into the square because TRIANGLE says RIGHT TURN.” A cure (one of several equally simple ones) is inherent in this diagnosis: Make a triangle procedure with left turns.
同样，如果一个成年人以为他可以通过重复[向前100向右 转60]让 Turtle 画出一个三角形，他会惊讶地看到六边形出现。但可以“进入”程序并了解为什么会发生这种情况。此外，可以进行反省，看看这个错误是如何从对欧几里得三角定理的最常见陈述的非常肤浅的理解中产生的：“三角形的角度总和是 180 度。”
Similarly an adult who thought he could make the Turtle draw a triangle by REPEAT [FORWARD 100 RIGHT TURN 60] would be astonished to see a hexagon appear. But it is possible to “get into” the program and see why this happens. Moreover, it is possible to introspect and see how the bug came from a very superficial understanding of the most common statement of Euclid’s triangle theorem: “The sum of the angles of a triangle is 180 degrees.”
孩子（事实上，也许大多数成年人也一样）生活在一个对一切事物都只有部分了解的世界：也许了解得足够多，但永远无法完全理解。对许多人来说，完全理解乌龟的行为，以至于再也无话可说，这是一种罕见的、可能是独一无二的体验。对某些人来说，这是一种令人振奋的经历：我们可以从孩子们渴望解释他们所理解的东西中看出这一点。对所有人来说，这都是大多数人从未遇到过的分析知识的更清晰的模型。
A child (and, indeed, perhaps most adults) lives in a world in which everything is only partially understood: well enough perhaps, but never completely. For many, understanding the Turtle’s action so completely that there is nothing more to say about it is a rare, possibly unique, experience. For some it is an exhilarating one: We can see this by the children’s eagerness to explain what they have understood. For all it is a better model of the crispness of analytic knowledge than most people ever encounter.
读者可能会反对说，儿童程序员根本无法“完全”理解 Turtle，他们几乎无法理解 Turtle 执行LOGO命令时幕后运作的复杂机制。我们是否真的有可能让儿童陷入一种复杂技术的环境中，而高级计算机科学家只能部分理解这种技术的复杂性？
The reader might object that far from understanding the Turtle “fully” a child programmer hardly understands at all the complex mechanisms at work behind the scenes whenever a Turtle carries out a LOGO command. Are we in fact in danger of mystifying children by placing them in an environment of sophisticated technology whose complexities are only partially understood by advanced computer scientists?
这些担忧让我们回到本章开头讨论的问题。例如，我提议用简单程序的形式描述杂耍。但同样的担忧也出现了：用程序语言进行的描述是否抓住了杂耍过程的本质，还是掩盖了杂耍的复杂性，使人产生困惑？
These concerns bring us back full circle to the issues with which this chapter began. For example, I proposed a description of juggling in the form of a simple program. But the same concern arises: Does the description in procedural language grasp the essence of the process of juggling, or does it mystify by covering over the complexities of the juggling?
这些问题非常笼统，涉及科学方法的基本问题。牛顿通过将整个星球简化为按照一组固定的运动定律移动的点来“理解”宇宙。这是抓住了现实世界的本质还是隐藏了它的复杂性？能够像科学家一样思考的部分含义是对这些认识论问题有直观的理解，我相信与海龟一起工作可以让孩子们有机会了解这些问题。
These questions are very general and touch on fundamental issues of scientific method. Newton “understood” the universe by reducing whole planets to points that move according to a fixed set of laws of motion. Is this grasping the essence of the real world or hiding its complexities? Part of what it means to be able to think like a scientist is to have an intuitive understanding of these epistemological issues, and I believe that working with Turtles can give children an opportunity to get to know them.
事实上，孩子们很容易理解“乌龟”如何定义一个自足的世界，在这个世界中，某些问题是相关的，而其他问题则不是。下一章讨论了如何通过构建许多这样的“微观世界”来发展这一想法，每个“微观世界”都有自己的一套假设和约束。孩子们会了解到，不受无关问题的干扰，探索选定的微观世界的属性是什么感觉。通过这样做，他们学会将探索习惯从个人生活转移到科学理论构建的正式领域。
It is in fact easy for children to understand how the Turtle defines a self-contained world in which certain questions are relevant and others are not. The next chapter discusses how this idea can be developed by constructing many such “microworlds,” each with its own set of assumptions and constraints. Children get to know what it is like to explore the properties of a chosen microworld undisturbed by extraneous questions. In doing so they learn to transfer habits of exploration from their personal lives to the formal domain of scientific theory construction.
计算机世界的内部可理解性为儿童提供了执行比物理世界通常更复杂的项目的机会。许多孩子想象着他们可以用拼装玩具建造复杂的结构，或者幻想着将他们的朋友组织起来开展复杂的企业。但当他们试图实现这样的项目时，他们很快就会遇到物质和人的难以理解的限制。由于计算机程序原则上可以完全按照预期的方式运行，因此可以更安全地将它们组合成复杂的系统。因此，孩子们能够获得对复杂性的感觉。
The internal intelligibility of computer worlds offers children the opportunity to carry out projects of greater complexity than is usually possible in the physical world. Many children imagine complex structures they might build with an erector set or fantasize about organizing their friends into complex enterprises. But when they try to realize such projects, they too soon run into the unintelligible limitations of matter and people. Because computer programs can in principle be made to behave exactly as they are intended to, they can be combined more safely into complex systems. Thus, children are able to acquire a feel for complexity.
现代科学和工程为实现直到最近才令人难以想象的复杂程度的项目创造了机会。但科学也教会了我们简单的力量，我以一个我认为很感人的故事结束本章，这个故事讲述了一个孩子在一个特别简单但对我个人来说很重要的经历中学到了一些这一点。
Modern science and engineering have created the opportunity for achieving projects of a degree of complexity scarcely imaginable until recently. But science teaches us the power of simplicity as well and I end the chapter with what I find to be a moving story of a child who learned something of this in a particularly simple but personally important experience.
黛博拉是一名六年级学生，在学习上存在一些困难。在学习过程中，她向孩子们展示了如何执行“前进”、“左转”和“右转”的命令，从而了解了屏幕上的乌龟世界。许多孩子发现，这些命令可以分配为任意数字，这是令人振奋的力量源泉和令人兴奋的探索领域。黛博拉对她在学校所做的大部分事情的反应感到恐惧。在她最初几个小时的乌龟训练中，她对教练产生了一种令人不安的依赖程度，在迈出最小的探索步骤之前，她总是要求教练给予安慰。当黛博拉决定限制她的乌龟命令时，转折点出现了，她在乌龟命令的微观世界中创造了一个微观世界。她只允许自己执行一个转动命令：向右30 度。要将乌龟转 90 度，她需要重复“向右30 度”三次，而要达到向左30 度的效果，则需要重复十一次。对于旁观者来说，用如此复杂的方法获得简单的效果似乎很乏味。但对于黛博拉来说，能够构建自己的微观世界并发现自己能在严格的限制下做多少事情是令人兴奋的。她不再寻求允许去探索。有一天，当老师提出向她展示一种“更简单的方法”来实现某种效果时，她耐心地听着，说：“我不认为我会那样做。”几周后，当她准备好时，她带着一种新的自信出现了，这种自信不仅体现在更雄心勃勃的海龟项目中，也体现在她与学校里所做的其他一切的关系中。我喜欢从黛博拉的经历中看到一个小小的重演，即哥白尼和伽利略等思想家的成功如何让人们摆脱与物理无关的迷信依赖。在这两种情况下——在黛博拉的个人历史和西方思想史上——数学理论的成功都不仅仅是一种工具性的作用：它肯定了思想的力量和心灵的力量。
Deborah, a sixth grader who had problems with school learning, was introduced to the world of screen Turtles by being shown how they could obey the commands FORWARD, LEFT, and RIGHT. Many children find the fact that these commands can be assigned any number an exhilarating source of power and an exciting area of exploration. Deborah found it frightening, the reaction she had to most of what she did at school. In her first few hours of Turtle work she developed a disturbing degree of dependence on the instructor, constantly asking for reassurance before taking the smallest exploratory step. A turning point came when Deborah decided to restrict her Turtle commands, creating a microworld within the microworld of Turtle commands. She allowed herself only one turning command: RIGHT 30. To turn the Turtle through 90 degrees, she would repeat RIGHT 30 three times and would obtain the effect of LEFT 30 by repeating it eleven times. To an onlooker it might seem tedious to obtain simple effects in such complicated ways. But for Deborah it was exciting to be able to construct her own microworld and to discover how much she could do within its rigid constraints. She no longer asked permission to explore. And one day, when the teacher offered to show her a “simpler way” to achieve an effect, she listened patiently and said, “I don’t think I’ll do it that way.” She emerged when she was ready, several weeks later, with a new sense of confidence that showed itself not only in more ambitious Turtle projects but in her relationship to everything else she did in school. I like to see in Deborah’s experience a small recapitulation of how the success of such thinkers as Copernicus and Galileo allowed people to break away from superstitious dependencies that had nothing in themselves to do with physics. In both cases—in Deborah’s personal history and in the history of Western thought—the success of a mathematical theory served more than an instrumental role: It served as an affirmation of the power of ideas and the power of the mind.
第五章
CHAPTER 5
微观世界
Microworlds
知识孵化器
Incubators for Knowledge
我 将数学之于学习定义为启发法之于解决问题：数学原理是阐明和促进学习过程的思想。在本章中，我们将重点介绍两个重要的数学原理，它们是大多数人的常识，告诉人们在遇到新玩意、新舞步、新想法或新词时该怎么做。首先，将新知识和要学习的知识与您已知的知识联系起来。其次，将新知识变成您自己的知识：用它创造新的东西，用它玩耍，用它构建。例如，要学习一个新词，我们首先要寻找熟悉的“词根”，然后通过在自己构造的句子中使用该词来练习。
I HAVE DEFINED MATHETICS AS BEING TO LEARNING AS HEURISTICS IS TO problem solving: Principles of mathetics are ideas that illuminate and facilitate the process of learning. In this chapter we focus on two important mathetic principles that are part of most people’s commonsense knowledge about what to do when confronted with a new gadget, a new dance step, a new idea, or a new word. First, relate what is new and to be learned to something you already know. Second, take what is new and make it your own: Make something new with it, play with it, build with it. So for example, to learn a new word, we first look for a familiar “root” and then practice by using the word in a sentence of our own construction.
我们在流行的、常识性的学习理论中发现了关于如何学习的两步格言：一代又一代的父母和老师向一代又一代的小学生传授了学习新单词的程序。它也与最早的学习过程中使用的策略相对应。皮亚杰研究了儿童的自发学习，发现这两个步骤都在起作用——儿童在皮亚杰称之为同化的过程中将新知识吸收到旧知识中，儿童在积极运用知识的过程中构建自己的知识。
We find this two-step dictum about how to learn in popular, commonsense theories of learning: The procedure described for learning a new word has been given to generations of elementary schoolchildren by generations of parents and teachers. And it also corresponds to the strategies used in the earliest processes of learning. Piaget has studied the spontaneous learning of children and found both steps at work—the child absorbs the new into the old in a process that Piaget calls assimilation, and the child constructs his knowledge in the course of actively working with it.
但这一过程中经常会遇到障碍。新知识往往与旧知识相矛盾，而有效的学习需要应对这种冲突的策略。有时可以调和相互冲突的知识，有时必须放弃其中之一，有时如果安全地保存在不同的心理隔间中，两者可以“保留”。我们将通过研究一个特殊案例来研究这些学习策略，在这个案例中，物理学的形式理论与常识、直觉的物理学观念发生了尖锐冲突。
But there are often roadblocks in the process. New knowledge often contradicts the old, and effective learning requires strategies to deal with such conflict. Sometimes the conflicting pieces of knowledge can be reconciled, sometimes one or the other must be abandoned, and sometimes the two can both be “kept around” if safely maintained in separate mental compartments. We shall look at these learning strategies by examining a particular case in which a formal theory of physics enters into sharp conflict with commonsense, intuitive ideas about physics.
最简单的冲突之一来自牛顿物理学的基本原理：运动中的物体如果不受干扰，将永远以恒定速度沿直线运动。这一“永动机”原理与常识相悖，甚至与亚里士多德等较古老的物理学理论相悖。
One of the simplest of such conflicts is raised by the fundamental tenet of Newton’s physics: A body in motion will, if left alone, continue to move forever at a constant speed and in a straight line. This principle of “perpetual motion” contradicts common experience and, indeed, older theories of physics such as Aristotle’s.
假设我们想移动一张桌子。我们施加一个力，让桌子运动起来，并继续施加力，直到桌子到达所需的位置。当我们停止推动时，桌子就会停止。从表面上看，桌子的行为并不像牛顿物体。教科书告诉我们，如果是这样，一次推动就会让它永远运动，需要一种反作用力才能让它停在所需的位置。
Suppose we want to move a table. We apply a force, set the table in motion, and keep on applying the force until the table reaches the desired position. When we stop pushing, the table stops. To our superficial gaze, the table does not behave like a Newtonian object. If it did, textbooks tell us, one push would set it in motion forever and a counteracting force would be needed to stop it at the desired place.
理想理论与日常观察之间的冲突只是学习牛顿物理学的几个障碍之一。其他障碍则源于应用这两个数学原理的困难。根据第一个原则，想要学习牛顿物理学的人应该找到将其与他们已知的事物联系起来的方法。但他们可能不具备任何可以有效与之相关的知识。根据第二个原则，一个好的学习策略是运用牛顿运动定律，以个性化和有趣的方式运用它们。但这也不是那么简单。除非你掌握了牛顿定律的方法和一些可以应用它们的熟悉材料，否则你无法用牛顿定律做任何事情。
This conflict of ideal theory and everyday observation is only one of several roadblocks to the learning of Newtonian physics. Others derive from difficulties in applying the two mathetic principles. According to the first, people who want to learn Newtonian physics should find ways to relate it to something they already know. But they may not possess any knowledge to which it can be effectively related. According to the second, a good strategy for learning would be to work with the Newtonian laws of motion, to use them in a personal and playful fashion. But this too is not so simple. One cannot do anything with Newton’s laws unless one has some way to grab hold of them and some familiar material to which they can be applied.
本章的主题是计算思想如何成为思考牛顿定律的材料。关键思想已经被预见。我们看到，当 Turtle 而不是点作为构建块时，形式几何学变得更容易理解。在这里，我们对牛顿所做的就是对欧几里得所做的。牛顿定律使用“粒子”的概念来陈述，粒子是一个数学上抽象的实体，它与点相似，没有大小，但除了位置之外还具有其他一些属性：它具有质量和速度，或者，如果有人喜欢将这两者合并，它具有动量。在本章中，我们扩展了 Turtle 的概念，以包括行为类似于牛顿粒子的实体以及我们已经遇到的类似于欧几里得点的实体。这些新的 Turtle（我们称之为 Dynaturtle）更具动态性，因为它们的状态除了包括前面讨论的几何 Turtle 的两个几何分量（位置和方向）之外，还包括两个速度分量。而状态的更多部分则需要更丰富的命令语言：TURTLE TALK经过扩展，允许我们告诉 Turtle 以给定的速度移动。除了理解物理之外，更丰富的TURTLE TALK还立即开辟了许多视角。除了模拟真实或发明的物理定律外，Dynaturtle 还可以被置于运动模式中，用于美学、奇特或好玩的目的。过于狭隘的物理老师可能会认为这一切都是浪费时间：真正的工作是理解物理。但我希望为不同的物理教育理念辩护。我相信学习物理包括将物理知识与非常多样化的个人知识联系起来。为此，我们应该允许学习者构建和使用物理学家可能拒绝承认为物理学的过渡系统。1
The theme of this chapter is how computational ideas can serve as material for thinking about Newton’s laws. The key idea has already been anticipated. We saw how formal geometry becomes more accessible when the Turtle instead of the point is taken as the building block. Here we do for Newton what we did for Euclid. Newton’s laws are stated using the concept of “a particle,” a mathematically abstract entity that is similar to a point in having no size but that does have some other properties besides position: It has mass and velocity or, if one prefers to merge these two, it has momentum. In this chapter we enlarge our concept of Turtle to include entities that behave like Newton’s particles as well as those we have already met that resemble Euclid’s points. These new Turtles, which we call Dynaturtles, are more dynamic in the sense that their state is taken to include two velocity components in addition to the two geometric components, position and heading, of the previously discussed geometry Turtles. And having more parts to the state leads to requiring a slightly richer command language: TURTLE TALK is extended to allow us to tell the Turtle to set itself moving with a given velocity. This richer TURTLE TALK immediately opens up many perspectives besides the understanding of physics. Dynaturtles can be put into patterns of motion for aesthetic, fanciful, or playful purposes in addition to simulating real or invented physical laws. The too narrowly focused physics teacher might see all this as a waste of time: The real job is to understand physics. But I wish to argue for a different philosophy of physics education. It is my belief that learning physics consists of bringing physics knowledge into contact with very diverse personal knowledge. And to do this we should allow the learner to construct and work with transitional systems that the physicist may refuse to recognize as physics.1
大多数物理课程与数学课程相似，它们迫使学习者进入分离的学习模式，并将“有趣”的材料推迟到大多数学生能够保持足够学习动力的阶段之后。物理学的强大思想和智力美感在不断学习“先决条件”的过程中消失了。学习牛顿物理学可以作为数学策略如何被阻塞和畅通的一个例子。我们将描述一条绕过障碍的牛顿新“学习路径”：一个基于计算机的交互式学习环境，其中先决条件内置于系统中，学习者可以成为自己学习的主动构建者。
Most physics curricula are similar to the math curriculum in that they force the learner into dissociated learning patterns and defer the “interesting” material past the point where most students can remain motivated enough to learn it. The powerful ideas and the intellectual aesthetic of physics is lost in the perpetual learning of “prerequisites.” The learning of Newtonian physics can be taken as an example of how mathetic strategies can become blocked and unblocked. We shall describe a new “learning path” to Newton that gets around the block: a computer-based interactive learning environment where the prerequisites are built into the system and where learners can become the active, constructing architects of their own learning.
让我们首先仔细研究一下先决条件问题。想要学习空气动力学的人可能会在看到大学目录中令人兴奋的课程描述后面的一系列先决条件（包括力学和流体动力学）后失去兴趣。如果一个人想了解莎士比亚，就找不到先决条件列表。似乎可以公平地假设，先决条件列表是教育工作者认为的进入知识领域的学习路径的表达。进入空气动力学的学习路径是数学的，正如我们在我们的文化中所看到的，数学知识被归类为“特殊”知识——只在为这种深奥的知识保留的特殊地方才会被提及。大多数儿童的非学术学习环境对数学发展几乎没有任何推动力。这意味着学校和大学必须沿着极其正式的学习路径来学习空气动力学知识。通向莎士比亚的道路同样复杂，但其基本构成要素是我们普遍文化的一部分：我们假设许多人将能够非正式地学习它们。我们将开发的物理微观世界，即我们基于计算机的数学世界的物理模拟，提供了一条通往牛顿运动定律的皮亚杰式学习路径，这一主题通常被认为是只有通过漫长的、正规化的学习路径才能达到的知识的典范。牛顿的运动思维是一套复杂且看似违反直觉的世界假设。从历史上看，它的发展需要很长时间。就个人发展而言，儿童与环境的互动使他形成了一套非常不同的个人运动信念，这些信念在很多方面更接近亚里士多德而不是牛顿的信念。毕竟，亚里士多德的运动思想与我们经验中最常见的情况相对应。试图发展牛顿运动思维的学生会遇到三种问题，而计算机微观世界可以帮助解决这些问题。首先，学生们几乎没有直接体验过纯牛顿运动。当然，他们有过一些。例如，当一辆汽车在结冰的道路上打滑时，它就变成了一个牛顿物体：它完全可以在没有外界帮助的情况下继续保持其运动状态。但司机的心理状态无法从学习经验中获益。在没有直接和身体体验的情况下，为了获得牛顿运动的经验，学校被迫让学生间接地、高度数学化地体验牛顿物体。他们通过操纵方程而不是操纵物体本身来学习运动。这种经验缺乏即时性，很难改变学生的直觉。而且它本身还需要其他正式的先决条件。学生必须先学会如何使用方程，然后才能用它们来模拟牛顿世界。我们的计算机微观世界可能提供帮助的最简单的方法是将学生置于一个模拟世界中，让他们可以直接接触牛顿运动。这可以在他们年轻的时候完成。不需要等到他们掌握方程。恰恰相反：它不是让学生等待方程，而是通过为他们提供直观易懂的使用环境来激励和促进他们掌握方程技能。
Let us begin with a closer look at the problem of prerequisites. Someone who wanted to learn about aerodynamics might lose interest upon seeing the set of prerequisites, including mechanics and hydrodynamics, that follow an exciting course description in a college catalogue. If one wants to learn about Shakespeare, one finds no list of prerequisites. It seems fair to assume that a list of prerequisites is an expression of what educators believe to be a learning path into a domain of knowledge. The learning path into aerodynamics is mathematical, and, as we have seen in our culture, mathematical knowledge is bracketed, treated as “special”—spoken of only in special places reserved for such esoteric knowledge. The nonacademic learning environments of most children provide little impetus to that mathematical development. This means that schools and colleges must approach the knowledge of aerodynamics along exceedingly formal learning paths. The route into Shakespeare is no less complex, but its essential constitutive elements are part of our general culture: It is assumed that many people will be able to learn them informally. The physics microworld we shall develop, the physics analog of our computer-based Mathland, offers a Piagetian learning path into Newtonian laws of motion, a topic usually considered paradigmatic of the kind of knowledge that can only be reached by a long, formalized learning path. Newtonian thinking about motion is a complex and seemingly counterintuitive set of assumptions about the world. Historically, it was long to evolve. And in terms of individual development, the child’s interaction with his environment leads him to a very different set of personal beliefs about motion, beliefs that in many ways are closer to Aristotle’s than to Newton’s. After all, the Aristotelian idea of motion corresponds to the most common situation in our experience. Students trying to develop Newtonian thinking about motion encounter three kinds of problems that a computer microworld could help solve. First, students have had almost no direct experience of pure Newtonian motion. Of course, they have had some. For example, when a car skids on an icy road it becomes a Newtonian object: It will, only too well, continue in its state of motion without outside help. But the driver is not in a state of mind to benefit from the learning experience. In the absence of direct and physical experiences of Newtonian motion, the schools are forced to give the student indirect and highly mathematical experiences of Newtonian objects. There movement is learned by manipulating equations rather than by manipulating the objects themselves. The experience, lacking immediacy, is slow to change the student’s intuitions. And it itself requires other formal prerequisites. The student must first learn how to work with equations before using them to model a Newtonian world. The simplest way in which our computer microworld might help is by putting students in a simulated world where they have direct access to Newtonian motion. This can be done when they are young. It need not wait for their mastery of equations. Quite the contrary: Instead of making students wait for equations, it can motivate and facilitate their acquisition of equational skills by providing an intuitively well-understood context for their use.
直接体验牛顿运动是学习牛顿物理学的宝贵财富。但要理解它，除了直观的、凭感觉的体验，还需要更多的东西。学生需要概念化和“捕捉”这个世界的方法。事实上，牛顿伟大贡献的核心部分是发明了一种形式主义，一种适合于此目的的数学。他称之为“流数”；今天的学生称之为“微分学”。计算机屏幕上的 Dynaturtle 允许初学者玩牛顿物体。Dynaturtle 的概念允许学生思考它们。而控制 Dynaturtle 行为的程序提供了一种形式主义，我们可以在其中捕捉我们原本转瞬即逝的想法。这样一来，它就绕过了漫长的路线（算术、代数、三角学、微积分），进入了从牛顿自己的著作到现代教科书只经过表面修改的形式主义。我相信它能让学生更深入地了解牛顿在开始写下方程式之前的思想。
Direct experience with Newtonian motion is a valuable asset for the learning of Newtonian physics. But more is needed to understand it than an intuitive, seat-of-the-pants experience. The student needs the means to conceptualize and “capture” this world. Indeed, a central part of Newton’s great contribution was the invention of a formalism, a mathematics suited to this end. He called it “fluxions”; present-day students call it “differential calculus.” The Dynaturtle on the computer screen allows the beginner to play with Newtonian objects. The concept of the Dynaturtle allows the student to think about them. And programs governing the behavior of Dynaturtles provide a formalism in which we can capture our otherwise too fleeting thoughts. In doing so it bypasses the long route (arithmetic, algebra, trigonometry, calculus) into the formalism that has passed with only superficial modification from Newton’s own writing to the modern textbook. And I believe it brings the student in closer touch with what Newton must have thought before he began writing equations.
第三个前提条件则更为微妙。我们很快将直接研究通常称为牛顿运动定律的表述。在此过程中，许多读者无疑会回想起“运动定律”一词所引起的不安感。那是什么样的东西？除了牛顿运动定律之外，还有哪些其他运动定律？很少有学生在第一次接触牛顿时能回答这些问题，我相信这在很大程度上解释了物理学对大多数学习者来说为何如此困难。学生在不知道某样东西是什么的情况下，无法将其变成自己的。因此，第三个前提条件是，我们必须找到方法，促进个人不仅掌握牛顿运动及其定律，而且掌握描述运动的一般定律。我们通过设计一系列微观世界来实现这一点。
The third prerequisite is somewhat more subtle. We shall soon look directly at statements of what is usually known as Newton’s laws of motion. As we do, many readers will no doubt recall a sense of unease evoked by the phrase “law of motion.” What kind of a thing is that? What other laws of motion are there besides Newton’s? Few students can answer these questions when they first encounter Newton, and I believe that this goes far toward explaining the difficulty of physics for most learners. Students cannot make a thing their own without knowing what kind of a thing it is. Therefore, the third prerequisite is that we must find ways to facilitate the personal appropriation not only of Newtonian motion and the laws that describe it but also of the general notion of laws that describe motion. We do this by designing a series of microworlds.
海龟世界是一个微观世界，一个“地方”，一个“数学王国”，某些数学思维可以在这里特别轻松地孵化和成长。微观世界是一个孵化器。现在，我们将设计一个微观世界作为牛顿物理学的孵化器。微观世界的设计使其成为特定种类的强大思想或智力结构的“成长场所”。因此，我们设计的微观世界不仅体现了“正确的”牛顿思想，还体现了许多其他思想：具有历史和心理学重要性的亚里士多德思想、更复杂的爱因斯坦思想，甚至是一个“广义运动定律世界”，它充当个人可以自己发明的无限多种运动定律的框架。因此，学习者可以通过任意数量的中间世界从亚里士多德进步到牛顿，甚至进步到爱因斯坦。在随后的描述中，牛顿所面临的数学障碍被克服了：先决条件植根于个人知识，学习者参与对思想和各种运动定律的创造性探索。
The Turtle World was a microworld, a “place,” a “province of Mathland,” where certain kinds of mathematical thinking could hatch and grow with particular ease. The microworld was an incubator. Now we shall design a microworld to serve as an incubator for Newtonian physics. The design of the microworld makes it a “growing place” for a specific species of powerful ideas or intellectual structures. So, we design microworlds that exemplify not only the “correct” Newtonian ideas, but many others as well: the historically and psychologically important Aristotelian ones, the more complex Einsteinian ones, and even a “generalized law-of-motion world” that acts as a framework for an infinite variety of laws of motion that individuals can invent for themselves. Thus learners can progress from Aristotle to Newton and even to Einstein via as many intermediate worlds as they wish. In the descriptions that follow, the mathetic obstacles to Newton are overcome: The prerequisites are rooted in personal knowledge and the learner is involved in a creative exploration of the idea and the variety of laws of motion.
让我们从牛顿三大定律开始描述微观世界，这里以“正式”的形式表述，读者不需要详细理解：
Let us begin to describe the microworld by starting with Newton’s three laws, stated here “formally” and in a form that readers do not have to understand in detail:
1. 每个粒子都会保持静止状态或以恒定速度沿直线运动，除非受到力的迫使而改变该状态。
1. Every particle continues in a state of rest or motion with constant speed in a straight line unless compelled by a force to change that state.
2. 产生运动变化的净不平衡力 ( F ) 等于质点的质量 ( m ) 与加速度 ( a ) 的乘积：F = ma 。
2. The net unbalanced force (F) producing a change of motion is equal to the product of the mass (m) and the acceleration (a) of the particle: F = ma.
3. 所有的力都来自粒子的相互作用，每当一个粒子作用于另一个粒子时，第一个粒子上就会产生大小相等、方向相反的反作用力。
3. All forces arise from the interaction of particles, and whenever a particle acts on another there is an equal and opposite reaction on the first.
正如我们所指出的，阻碍儿童理解这些定律的不仅仅是用来表述这些定律的深奥语言。我们分析这些障碍是为了推断出我们微观世界的设计标准。第一个障碍是儿童不知道任何其他类似这些定律的东西。在接受牛顿运动定律之前，他们应该知道一些其他运动定律。必须有一个运动定律的第一个例子，但它肯定不必像牛顿定律那样复杂、微妙和违反直觉。更明智的做法是让学习者通过研究一个非常简单易懂的运动定律实例来获得运动定律的概念。这将是我们微观世界的第一个设计标准。第二个障碍是，正如所提到的，这些定律没有为想要操纵它们的学习者提供立足点。除了课本章末的练习之外，他们没有别的用处。因此，我们微观世界的第二个设计标准是活动、游戏、艺术等的可能性，这些活动使微观世界中的活动变得重要。第三个障碍是牛顿定律使用了许多超出大多数人经验的概念，例如“状态”的概念。我们的微观世界将被设计成所有需要的概念都可以在那个世界的经验中定义。
As we have noted, children’s access to these laws is blocked by more than the recondite language used to state them. We analyze these roadblocks in order to infer design criteria for our microworld. A first block is that children do not know anything else like these laws. Before being receptive to Newton’s laws of motion, they should know some other laws of motion. There must be a first example of laws of motion, but it certainly does not have to be as complex, subtle, and counterintuitive as Newton’s laws. More sensible is to let the learner acquire the concept of laws of motion by working with a very simple and accessible instance of a law of motion. This will be the first design criterion for our microworld. The second block is that the laws, as stated, offer no footholds for learners who want to manipulate them. There is no use they can put them to outside of end-of-chapter schoolbook exercises. And so, a second design criterion for our microworlds is the possibility of activities, games, art, and so on that make activity in the microworlds matter. A third block is the fact that the Newtonian laws use a number of concepts that are outside most people’s experience, the concept of “state,” for example. Our microworld will be designed so that all needed concepts can be defined within the experience of that world.
和几何乌龟一样，物理乌龟也是一个可交互的存在，学习者可以操纵它，从而为主动学习提供环境。但学习的“主动”性不仅仅体现在交互的意义上。物理微观世界中的学习者能够发明他们自己对微观世界及其法则的假设，并能够使它们成为现实。他们可以塑造他们一天的工作现实，他们可以修改它并构建替代方案。这是一种有效的学习方式，与我们每个人曾经进行的一些最有效的学习方式相似。皮亚杰已经证明，儿童通过首先构建他们自己的、非常不同的（例如，前保护主义者）数学来学习基本的数学思想。儿童通过首先学习他们自己的（“儿语”）方言来学习语言。因此，当我们将微观世界视为强大创意的孵化器时，我们试图借鉴这种有效的策略：我们允许学习者自由地发明许多可以在尽可能多的发明世界中发挥作用的东西，从而让他们学习“官方”物理学。
As in the case of the geometry Turtle, the physics Turtle is an interactive being that can be manipulated by the learner, providing an environment for active learning. But the learning is not “active” simply in the sense of interactive. Learners in a physics microworld are able to invent their own personal sets of assumptions about the microworld and its laws and are able to make them come true. They can shape the reality in which they will work for the day, they can modify it and build alternatives. This is an effective way to learn, paralleling the way in which each of us once did some of our most effective learning. Piaget has demonstrated that children learn fundamental mathematical ideas by first building their own, very much different (for example, preconservationist) mathematics. And children learn language by first learning their own (“baby-talk”) dialects. So, when we think of microworlds as incubators for powerful ideas, we are trying to draw upon this effective strategy: We allow learners to learn the “official” physics by allowing them the freedom to invent many that will work in as many invented worlds.
遵循波利亚将新事物与旧事物联系起来理解的原则，让我们将海龟几何的微观世界重新解释为一种特殊物理学的微观世界。我们以与牛顿定律平行的形式重新制定海龟运动的定律。这给了我们以下“海龟运动定律”。当然，在只有一个海龟的世界中，处理粒子之间相互作用的第三定律将没有类似物。
Following Polya’s principle of understanding the new by associating it with the old, let us reinterpret our microworld of Turtle geometry as a microworld of a special kind of physics. We recast the laws by which Turtles work in a form that parallels the Newtonian laws. This gives us the following “Turtle laws of motion.” Of course, in a world with only one Turtle, the third law, which deals with the interaction among particles, will not have an analog.
1. 每个海龟都保持静止状态，直到受到海龟命令而被迫改变该状态。
1. Every Turtle remains in its state of rest until compelled by a TURTLE COMMAND to change that state.
2. a. 命令FORWARD的输入等于 Turtle状态中POSITION部分的改变。
2. a. The input to the command FORWARD is equal to the Turtle’s change in the POSITION part of its state.
b. 命令RIGHT TURN的输入等于 Turtle 改变其状态的HEADING部分。
b. The input to the command RIGHT TURN is equal to the Turtle’s change of the HEADING part of its state.
通过这个练习，我们对牛顿物理学的理解得到了哪些提升？了解 Turtle 几何（并因此能够认识到 Turtle 运动定律中对它的重述）的学生现在如何看待牛顿定律？他们能够通过将牛顿前两个定律与他们已知的事物进行比较，以定性和直观的形式阐述牛顿前两个定律的实质。他们了解状态和状态变化运算符。在 Turtle 世界中，状态的两个组成部分各有一个状态变化运算符。运算符FORWARD可改变位置。运算符TURN可改变航向。在物理学中，只有一种状态变化运算符，称为力。力的作用是改变速度（或者更准确地说是动量）。位置会自行改变。
What have we gained in our understanding of Newtonian physics by this exercise? How can students who know Turtle geometry (and can thus recognize its restatement in Turtle laws of motion) now look at Newton’s laws? They are in a position to formulate in a qualitative and intuitive form the substance of Newton’s first two laws by comparing them with something they already know. They know about states and state-change operators. In the Turtle world, there is a state-change operator for each of the two components of the state. The operator FORWARD changes the position. The operator TURN changes the heading. In physics, there is only one state-change operator, called force. The effect of force is to change velocity (or, more precisely, momentum). Position changes by itself.
这些对比使学生对牛顿有了定性的理解。尽管海龟定律和牛顿运动定律之间仍然存在差距，但孩子们可以通过理解第一个定律来理解第二个定律。这样的孩子在学习物理方面已经领先了一大步。但我们可以做更多的事情来缩小海龟世界和牛顿世界之间的差距。我们可以设计其他海龟微观世界，在这些微观世界中，运动定律会更接近牛顿的情况。
These contrasts lead students to a qualitative understanding of Newton. Although there remains a gap between the Turtle laws and the Newtonian laws of motion, children can appreciate the second through an understanding of the first. Such children are already a big step ahead in learning physics. But we can do more to close the gap between Turtle and Newtonian worlds. We can design other Turtle microworlds in which the laws of motion move toward a closer approximation of the Newtonian situation.
为此，我们创建了一类 Turtle 微世界，其构成 Turtle 状态的属性和改变这些状态的运算符有所不同。我们正式描述了几何 Turtle，说它的状态由位置和速度组成，并且其状态改变运算符独立于这两个组件起作用。但是还有另一种方式，也许是更强大和直观的方式来思考它。那就是将 Turtle 视为“理解”某些通信类型而不理解其他通信类型的存在。因此，几何 Turtle 理解改变其位置同时保持其航向的命令，以及改变其航向同时保持其位置的命令。本着同样的精神，我们可以将牛顿 Turtle 定义为只能接受一种命令的存在，即会改变其动量的命令。这些描述实际上是我们在向儿童介绍微世界时使用的描述。现在让我们来看看两个可以说介于几何 Turtle 和牛顿 Turtle 之间的 Turtle 微世界。
To do this we create a class of Turtle microworlds that differs in the properties that constitute the state of the Turtle and in the operators that change these states. We have formally described the geometry Turtle by saying that its state consists of position and velocity and that its state-change operators act independently of these two components. But there is another way, perhaps a more powerful and intuitive way, to think about it. This is to see the Turtle as a being that “understands” certain kinds of communication and not others. So, the geometry Turtle understood the command to change its position while keeping its heading and to change its heading while keeping its position. In the same spirit, we could define a Newtonian Turtle as a being that can accept only one kind of order, one that will change its momentum. These kinds of description are in fact the ones we use in introducing children to microworlds. Now let us turn to two Turtle microworlds that can be said to lie between the geometry and Newtonian Turtles.
速度龟
VELOCITY TURTLES
速度 Turtle 的状态是POSITION 和 VELOCITY。当然，由于速度定义为位置的变化，根据定义，此状态的第一个分量是连续变化的（除非VELOCITY为零）。因此，为了控制速度 Turtle，我们只需告诉它采用什么速度。我们通过一个状态更改运算符（称为SETVELOCITY的命令）来执行此操作。
The state of a velocity Turtle is POSITION AND VELOCITY. Of course, since velocity is defined as a change in position, by definition the first component of this state is continuously changing (unless VELOCITY is zero). So, in order to control a velocity Turtle, we only have to tell it what velocity to adopt. We do this by one state-change operator, a command called SETVELOCITY.
加速海龟
ACCELERATION TURTLES
另一个 Turtle 介于几何 Turtle 和可以表示牛顿粒子的 Turtle 之间，即加速度 Turtle。在这里，Turtle 的状态也是其位置和速度。但这次 Turtle 无法理解诸如“以某某速度”之类的命令。它只能接受“将速度改变x，无论速度是多少”形式的指令。这个 Turtle 的行为就像一个质量不变的牛顿粒子。
Another Turtle, intermediate between the geometry Turtle and the Turtle that could represent a Newtonian particle, is an acceleration Turtle. Here, too, the state of the Turtle is its position and velocity. But this time the Turtle cannot understand such a command as “Take on such-and-such a velocity”. It can only take instructions of the form “Change your velocity by x, no matter what your velocity happens to be.” This Turtle behaves like a Newtonian particle with an unchangeable mass.
因此，乌龟的顺序——从几何乌龟到速度乌龟到加速度乌龟到牛顿乌龟——构成了一条通往牛顿的道路，这条道路与我们的两个数学原理相呼应。每一步都以清晰透明的方式建立在前一步的基础上，满足了先决条件原则。至于我们的第二个数学原理——“使用它，玩它”——情况更加引人注目。皮亚杰向我们展示了儿童如何利用他环境中的材料（触觉、视觉和动觉）构建一个前保护主义世界，然后是一个保护主义世界。但在计算机出现之前，只有非常贫乏的环境材料可用于构建牛顿世界。然而，我们描述的每个微观世界都可以作为一个可探索和可操纵的环境。
Thus, the sequence of turtles—geometry Turtle to velocity Turtle to acceleration Turtle to Newtonian Turtle—constitutes a path into Newton that is resonant with our two mathetic principles. Each step builds on the one before in a clear and transparent way, satisfying the principle of prerequisites. As for our second mathetic principle—“use it, play with it”—the case is even more dramatic. Piaget showed us how the child constructs a preconservationist and then a conservationist world out of the materials (tactile, visual, and kinesthetic) in his environment. But until the advent of the computer, there were only very poor environmental materials for the construction of a Newtonian world. However, each of the microworlds we described can function as an explorable and manipulable environment.
在海龟几何中，几何是通过计算机图形项目来教授的，这些项目产生的效果与本书插图中所示的效果类似。海龟几何中的每个新想法都为行动开辟了新的可能性，因此可以将其视为个人力量的源泉。使用诸如SETVELOCITY和CHANGE VELOCITY之类的新命令，学习者可以使事物运动起来，并产生形状和大小不断变化的设计。他们现在拥有了更大的个人力量和“拥有”动态的感觉。他们可以制作计算机动画——他们与电视或弹球馆中看到的东西建立了一种新的个人关系。电视节目、动画片或视频游戏的动态视觉效果现在鼓励他们问自己如何制作他们所看到的东西。这是一种不同于学生传统上在“科学实验室”中回答的问题。在传统的实验室教学法中，孩子们的任务是确定一个既定的真理。孩子们最多只能学到“这就是世界运作的方式”。在这些动态的 Turtle 微观世界中，他们会获得不同的理解——感受到世界为什么会这样运转。通过尝试许多不同的运动定律，孩子们会发现牛顿定律确实是移动物体最经济、最优雅的定律。
In Turtle geometry, geometry was taught by way of computer graphics projects that produce effects like those shown in the designs illustrating this book. Each new idea in Turtle geometry opened new possibilities for action and could therefore be experienced as a source of personal power. With new commands such as SETVELOCITY and CHANGE VELOCITY, learners can set things in motion and produce designs of ever-changing shapes and sizes. They now have even more personal power and a sense of “owning” dynamics. They can do computer animation—there is a new, personal relationship to what they see on television or in a pinball gallery. The dynamic visual effects of a TV show, an animated cartoon, or a video game now encourage them to ask how they could make what they see. This is a different kind of question than the one students traditionally answer in their “science laboratory.” In the traditional laboratory pedagogy, the task posed to the children is to establish a given truth. At best, children learn that “this is the way the world works.” In these dynamic Turtle microworlds, they come to a different kind of understanding—a feel for why the world works as it does. By trying many different laws of motion, children will find that the Newtonian ones are indeed the most economical and elegant for moving objects around.
前面的讨论都是关于牛顿的前两个定律。在 Turtles 世界中，牛顿第三定律的类似物有哪些？第三定律只有在交互实体的微观世界中才有意义 — — 对牛顿来说是粒子，对我们来说是 Turtles。因此，让我们假设一个有许多 Turtles 的微观世界，我们将其称为TURTLE 1、TURTLE 2 等等。如果我们给每个 Turtles 命名，我们就可以使用TURTLE TALK与多个 Turtles 进行通信。因此，我们可以使用如下命令：TELL TURTLE 4 SETVELOCITY 20（意思是“告诉 4 号 Turtle 以 20 的速度移动”。）
All of the preceding discussion has dealt with Newton’s first two laws. What analogs to Newton’s third law are possible in the world of Turtles? The third law is only meaningful in a microworld of interactive entities—particles for Newton, Turtles for us. So let us assume a microworld with many Turtles that we shall call TURTLE 1, TURTLE 2, and so on. We can use TURTLE TALK to communicate with multiple Turtles if we give each of them a name. So we can use commands such as: TELL TURTLE 4 SETVELOCITY 20 (meaning “Tell Turtle number 4 to take on a velocity of 20).
牛顿第三定律表达了一种宇宙模型，一种将物理现实的运作概念化为自我延续机器的方法。在这种宇宙观中，所有行为都由粒子相互施加力来控制，不受任何外部因素的干预。为了在 Turtle 微观世界中对此进行建模，我们需要许多相互交互的 Turtle。在这里，我们将开发两个用于思考交互的 Turtle 的模型：链接的 Turtle 和链接的 Dynaturtle。
Newton’s third law expresses a model of the universe, a way to conceptualize the workings of physical reality as a self-perpetuating machine. In this vision of the universe, all actions are governed by particles exerting forces on one another, with no intervention by any outside agent. In order to model this in a Turtle microworld, we need many Turtles interacting with each other. Here we shall develop two models for thinking about interacting Turtles: linked Turtles and linked Dynaturtles.
在第一个模型中，我们认为海龟是相互发出命令，而不是服从外部命令。它们是链接的海龟。当然，海龟可以通过多种方式链接。我们可以制作直接模拟牛顿粒子的海龟，并通过模拟重力链接起来。这通常在LOGO实验室中进行，在那里，通常被认为是大学物理中很难的主题被转化为初中生可以理解的形式。这种模拟可以作为从基本掌握牛顿力学到理解行星运动和航天器导航的跳板。他们通过使使用牛顿原理成为一个积极且个人参与的过程来实现这一点。但是要“掌握”相互作用粒子（或“链接的海龟”）的想法，学习者需要做更多。在给定的一组相互作用中工作永远不够。学习者需要知道不止一个相互作用定律的例子，并且应该有发明新定律的经验。链接海龟还有哪些例子？
In the first model we think of the Turtles as giving commands to one another rather than obeying commands from the outside. They are linked Turtles. Of course, Turtles can be linked in many ways. We can make Turtles that directly simulate Newtonian particles linked by simulated gravity. This is commonly done in LOGO laboratories, where topics usually considered difficult in college physics are translated into a form accessible to junior high school students. Such simulations can serve as a springboard from an elementary grasp of Newtonian mechanics to an understanding of the motion of planets and of the guidance of spacecraft. They do this by making working with the Newtonian principles an active and personally involving process. But to “own” the idea of interacting particles—or “linked Turtles”—the learner needs to do more. It is never enough to work within a given set of interactions. The learner needs to know more than one example of laws of interaction and should have experience inventing new ones. What are some other examples of linked Turtles?
第一个是“镜像海龟”微世界，里面有互相连接的海龟。我们从一个“镜像海龟”微世界开始，里面有两个海龟，它们按照以下规则连接在一起：只要给其中一个海龟一个“前进” （或“后退”）命令，另一个海龟也会执行相同的命令；只要给其中一个海龟一个“ 右转” （或“左转 ”）命令，另一个海龟也会执行相反的命令。这意味着，如果两个海龟一开始面对面，任何海龟程序都会使它们的行程成为彼此的镜像。一旦学习者理解了这一原理，就可以轻松制作出精美的万花筒设计。
A first is a microworld of linked Turtles called “mirror Turtles.” We begin with a “mirror Turtle” microworld containing two Turtles linked by the rules: Whenever either is given a FORWARD (or BACK) command, the other does the same; whenever either is given a RIGHT TURN (or LEFT TURN) command, the other does the opposite. This means that if the two Turtles start off facing one another, any Turtle program will cause their trips to be mirror images of one another’s. Once the learner understands this principle, attractive kaleidoscope designs can easily be made.
第二个由相连的海龟组成的微世界更接近牛顿物理学，它将这些镜像连接应用于速度海龟。本页上印刷的任何静态图像都无法传达这些动态万花筒的视觉刺激，在这些万花筒中，色彩鲜艳的光点在不断变化和旋转的路径中翩翩起舞。最终产品具有艺术的刺激性，但制作它的过程需要学习从相连的移动物体的动作和反应的角度进行思考。
A second microworld of linked Turtles, and one that is closer to Newtonian physics, applies these mirror linkages to velocity Turtles. No static images printed on this page could convey the visual excitement of these dynamic kaleidoscopes in which brightly colored points of light dance in changing and rotating paths. The end product has the excitement of art, but the process of making it involves learning to think in terms of the actions and reactions of linked moving objects.
这些相互关联的海龟微观世界巩固了学习者对三条运动定律的体验。但我们已经断言，多个微观世界也为理解运动定律的概念提供了一个平台。掌握了运动定律一般概念的学生拥有了一个新的、强大的解决问题的工具。让我们用猴子问题来说明一下。
These linked Turtle microworlds consolidate the learner’s experience of the three laws of motion. But we have asserted that multiple microworlds also provide a platform for understanding the idea of a law of motion. A student who has mastered the general concept of a law of motion has a new, powerful tool for problem solving. Let’s illustrate with the Monkey Problem.
一只猴子和一块石头分别绑在一根挂在滑轮上的绳子的两端。猴子和石头的重量相等，彼此保持平衡。猴子开始爬绳子。石头会怎样呢？
A monkey and a rock are attached to opposite ends of a rope that is hung over a pulley. The monkey and the rock are of equal weight and balance one another. The monkey begins to climb the rope. What happens to the rock?
我曾向数百名麻省理工学院的学生介绍过这个问题，他们全都成功通过了严格而全面的入门物理课程。超过四分之三的学生之前没有见过这个问题，他们给出了错误答案，或者无法决定如何解决它。有些人认为，石头的位置不会受到猴子攀爬的影响，因为无论猴子是否攀爬，它的质量都是相同的；有些人认为，石头会下降，要么是因为能量守恒，要么是因为与杠杆的类比；有些人猜测它会上升，但不知道为什么。这个问题显然“很难”。但这并不意味着它“复杂”。我认为，它的困难可以用缺少一些相当简单的东西来解释。当他们处理这个问题时，学生们会问自己：“这是一个‘能量守恒’问题吗？”“这是一个‘杠杆臂’问题吗？”等等。他们不会问自己：“这是一个‘运动定律’问题吗？”他们不会从这样的范畴来思考。在大多数学生的思维世界中，守恒定律、能量、杠杆臂等概念已成为思考的工具。它们是组织思维和解决问题的强大思想。对于在“运动定律”微观世界中有过经验的学生来说，“运动定律”也是如此。因此，这个学生不会因为猴子问题而无法提出正确的问题。这是一个运动定律问题，但一个只从代数公式的角度看待运动定律的学生甚至不会提出这个问题。对于那些提出这个问题的人来说，答案很容易得到。一旦人们将猴子和岩石视为相互关联的物体，类似于我们在海龟微观世界中研究的物体，很明显它们都必须经历相同的状态变化。由于它们以相同的速度（即零）开始，因此它们必须始终具有相同的速度。因此，如果一个上升，另一个也会以相同的速度上升。2
I have presented this problem to several hundred MIT students, all of whom had successfully passed rigorous and comprehensive introductory physics courses. Over three quarters of those who had not seen the problem before gave incorrect answers or were unable to decide how to go about solving it. Some thought the position of the rock would not be affected by the monkey’s climbing because the monkey’s mass is the same whether he is climbing or not; some thought that the rock would descend either because of a conservation of energy or because of an analogy with levers; some guessed it would go up, but did not know why. The problem is clearly “hard.” But this does not mean that it is “complex.” I suggest that its difficulty is explicable by the lack of something quite simple. When they approach the problem, students ask themselves: “Is this a ‘conservation-of-energy’ problem?” “Is this a ‘lever-arm’ problem?” and so on. They do not ask themselves: “Is this a ‘law-of-motion’ problem?” They do not think in terms of such a category. In the mental worlds of most students, the concepts of conservation, energy, lever-arm, and so on, have become tools to think with. They are powerful ideas that organize thinking and problem solving. For a student who has had experience in a “laws-of-motion” microworld, this is true of “law of motion.” Thus this student will not be blocked from asking the right question about the monkey problem. It is a law-of-motion problem, but a student who sees laws of motion only in terms of algebraic formulas will not even ask the question. For those who pose the question, the answer comes easily. And once one thinks of the monkey and the rock as linked objects, similar to the ones we worked with in the Turtle microworld, it is obvious that they must both undergo the same changes in state. Since they start with the same velocity, namely zero, they must therefore always have the same velocity. Thus, if one goes up, the other goes up at the same speed.2
我们提出微观世界是为了解决知识结构中出现的教学问题：先决条件问题。但微观世界也是对另一种问题的回应，这种问题并不根植于知识，而是根植于个人。这个问题与为“错误”（或者更确切地说是“过渡性”）理论的构建找到背景有关。我们所有人都通过构建、探索和理论构建来学习，但我们最初接触的大多数理论构建都产生了我们后来不得不放弃的理论。作为保守主义儿童，我们学会了如何构建和使用理论，只是因为我们被允许多年来持有关于数量的“离经叛道”观点。孩子们不会遵循从一个“正确立场”到另一个更高级的“正确立场”的学习路径。他们的自然学习路径包括“错误理论”，这些理论教授的理论构建与正确理论一样多。但在学校里，错误理论不再被容忍。
We have presented microworlds as a response to a pedagogical problem that arises from the structure of knowledge: the problem of prerequisites. But microworlds are a response to another sort of problem as well, one that is not embedded in knowledge but in the individual. The problem has to do with finding a context for the construction of “wrong” (or, rather, “transitional”) theories. All of us learn by constructing, exploring, and theory building, but most of the theory building on which we cut our teeth resulted in theories we would have to give up later. As preconservationist children, we learned how to build and use theories only because we were allowed to hold “deviant” views about quantities for many years. Children do not follow a learning path that goes from one “true position” to another, more advanced “true position.” Their natural learning paths include “false theories” that teach as much about theory building as true ones. But in school, false theories are no longer tolerated.
我们的教育系统拒绝接受儿童的“错误理论”，从而拒绝接受儿童真正的学习方式。它还拒绝接受指出错误理论学习路径重要性的发现。皮亚杰已经表明，儿童持有错误理论是学习思考过程的必要组成部分。幼儿的非正统理论不是缺陷或认知差距，而是锻炼认知能力、发展和运用更正统理论所需的必要技能的方式。教育工作者曲解了皮亚杰的思想，认为他的贡献揭示了儿童持有错误信念，而他们，教育工作者，必须克服这些错误信念。这使得皮亚杰在学校成为一种落后的皮亚杰——落后是因为在儿童准备好发明“正确”理论之前，他们就被强行灌输这些理论。落后是因为皮亚杰的工作质疑了“正确”理论作为一种学习策略更优越的观点。
Our educational system rejects the “false theories” of children, thereby rejecting the way children really learn. And it also rejects discoveries that point to the importance of the false-theory learning path. Piaget has shown that children hold false theories as a necessary part of the process of learning to think. The unorthodox theories of young children are not deficiencies or cognitive gaps, they serve as ways of flexing cognitive muscles, of developing and working through the necessary skills needed for more orthodox theorizing. Educators distort Piaget’s message by seeing his contribution as revealing that children hold false beliefs, which they, the educators, must overcome. This makes Piaget-in-the-schools a Piaget backward—backward because children are being force-fed “correct” theories before they are ready to invent them. And backward because Piaget’s work puts into question the idea that the “correct” theory is superior as a learning strategy.
有些读者可能难以将儿童的非自然保护主义世界观视为一种理论构建。让我们再举一个例子。皮亚杰问学龄前儿童：“是什么产生了风？”很少有人说“我不知道”。大多数孩子都给出了自己的理论，例如“树木通过摇动树枝产生了风”。这个理论虽然是错误的，但却为高度发达的理论构建技能提供了很好的证据。它可以用经验事实来检验。事实上，风的存在和树枝的摇动之间存在很强的相关性。孩子们可以进行实验，使他们的因果关系变得相当合理。当他们在脸附近挥动双手时，会产生非常明显的微风。孩子们可以想象，当挥动的物体不是一只小手而是一棵大树时，这种效果会成倍增加，并且当挥动的不是一棵而是许多棵大树时。所以，茂密森林中的树木应该是一个真正强大的风力发电机。
Some readers may have difficulty seeing the child’s nonconservationist view of the world as a kind of theory building. Let’s take another example. Piaget asked preschool children, “What makes the wind?” Very few said, “I don’t know.” Most children gave their own personal theories, such as, “The trees made the wind by waving their branches.” This theory, although wrong, gives good evidence for highly developed skills in theory building. It can be tested against empirical fact. Indeed there is a strong correlation between the presence of wind and the waving of tree branches. And children can perform an experiment that makes their causal connection quite plausible. When they wave their hands near their faces, they make a very noticeable breeze. Children can imagine this effect multiplied when the waving object is not a small hand but a giant tree, and when not one but many giant trees are waving. So, the trees of a dense forest should be a truly powerful wind generator.
对于一个提出如此美妙理论的孩子，我们该说什么呢？“约翰，你的想法很棒，但这个理论是错误的”这样的贬低会让大多数孩子相信自己提出理论是徒劳的。因此，解决办法是创造一个智力环境，让孩子不像学校那样被真假标准所主导，而不是扼杀他们的创造力。
What do we say to a child who has made such a beautiful theory? “That’s great thinking, Johnny, but the theory is wrong” constitutes a put-down that will convince most children that making one’s own theories is futile. So, rather than stifling the children’s creativity, the solution is to create an intellectual environment less dominated than the school’s by the criteria of true and false.
我们已经看到，微观世界就是这样的环境。就像喜欢使用具有第三定律相互作用的牛顿海龟进行编程的学生正在将牛顿变成自己的一样，在非牛顿微观世界中制作壮观螺旋的孩子也同样坚定地走在理解牛顿的道路上。他们都在学习如何使用变量，如何以不同质量的比率来思考，如何进行适当的近似，等等。他们在一个不是以真假和对错为决定标准的环境中学习数学和科学。
We have seen that microworlds are such environments. Just as students who prefer to do their programming using Newtonian Turtles with third law interaction are making Newton their own, children making a spectacular spiral in a non-Newtonian microworld are no less firmly on the path toward understanding Newton. Both are learning what it is like to work with variables, to think in terms of ratios of dissimilar qualities, to make appropriate approximations, and so on. They are learning mathematics and science in an environment where true or false and right or wrong are not the decisive criteria.
就像在一堂好的艺术课上一样，孩子学习技术知识是为了达到创造性和个人定义的目标。最终会有一个产品。老师和孩子都会为此感到兴奋。在算术课上，老师对孩子的成就表现出的快乐是真实的，但很难想象老师和孩子会因为一个产品而感到高兴。在LOGO环境中，这种情况经常发生。海龟微观世界中的螺旋是孩子创造的新颖而令人兴奋的作品——他甚至可能“发明”了它所基于的海龟连接方式。
As in a good art class, the child is learning technical knowledge as a means to get to a creative and personally defined end. There will be a product. And the teacher as well as the child can be genuinely excited by it. In the arithmetic class the pleasure that the teacher shows at the child’s achievement is genuine, but it is hard to imagine teacher and child showing delight over a product. In the LOGO environment it happens often. The spiral made in the Turtle microworld is a new and exciting creation by the child—he may even have “invented” the way of linking Turtles on which it is based.
老师对成果的真正兴奋传达给了孩子们，他们知道自己正在做一件有意义的事情。与算术课不同，在算术课上，他们知道自己做的加法只是练习，而在这里，他们可以认真对待自己的工作。如果他们刚刚通过命令乌龟向前迈出一系列长距离短距离并向右转几圈来画出一个圆，他们就准备与老师争论圆实际上是一个多边形。在五年级的LOGO课上听到过这种讨论的人，没有一个人不被这样的想法所打动：理论的真假与其对学习的贡献相比是次要的。
The teacher’s genuine excitement about the product is communicated to children who know they are doing something consequential. And unlike in the arithmetic class, where they know that the sums they are doing are just exercises, here they can take their work seriously. If they have just produced a circle by commanding the Turtle to take a long series of short forward steps and small right turns, they are prepared to argue with a teacher that a circle is really a polygon. No one who has overheard such a discussion in fifth-grade LOGO classes walks away without being impressed by the idea that the truth or falsity of theory is secondary to what it contributes to learning.
第六章
CHAPTER 6
头脑大小的强大想法
Powerful Ideas in Mind-Size Bites
“我喜欢你的微观世界，但它是物理学吗？
“I love your microworlds but is it physics?
“我不是说不是。但我该如何决定呢？”
I don’t say it is not. But how can I decide?”
—一位老师
—A TEACHER
两种认知方式之间的共同区别通常表现为“知道什么”与“知道怎样做”，或“命题知识”与“程序知识”，或“事实”与“技能”。在本章中，我们将讨论许多不能归结为这种二分法的任一术语的认知类型。日常生活中的重要例子是了解一个人、了解一个地方以及了解自己的心理状态。为了实现我们的主题，即使用计算机来理解植根于个人认知的科学认知，我们接下来将研究科学知识与了解一个人的相似之处，而不是了解一个事实或拥有一项技能。在此，我们将做一些类似于我们使用 Turtle 在形式几何和儿童身体几何之间建立桥梁的事情。在这里，我们的目标也是设计比传统学校所青睐的更协调的学习条件。在前面的章节中，我们探讨了一个悖论：尽管我们社会中的大多数人将数学归类为最难获得的知识，但矛盾的是，数学却是儿童最容易获得的知识。在本章中，我们将在科学领域遇到类似的悖论。我们将研究儿童思维与“真正的科学”之间的共同点，而不是“学校科学”与儿童或科学家思维之间的共同点。我们将再次注意到计算机进入和影响这种状况的方式的双重悖论。计算机的引入可以为悖论提供一种解决方法，但它通常会以加剧悖论的方式使用，因为它强化了对知识、对“学校数学”和“学校科学”的矛盾思维方式。
A COMMON DISTINCTION BETWEEN TWO WAYS OF KNOWING IS OFTEN expressed as “knowing-that” versus “knowing-how” or as “propositional knowledge” versus “procedural knowledge” or again as “facts” versus “skills.” In this chapter we talk about some of the many kinds of knowing that cannot be reduced to either term of this dichotomy. Important examples from everyday life are knowing a person, knowing a place, and knowing one’s own states of mind. In pursuit of our theme of using the computer to understand scientific knowing as rooted in personal knowing, we shall next look at ways in which scientific knowledge is more similar to knowing a person than similar to knowing a fact or having a skill. In this, we shall be doing something similar to how we used the Turtle to build bridges between formal geometry and the body geometry of the child. Here, too, our goal is to design conditions for more syntonic kinds of learning than those favored by the traditional schools. In previous chapters we have explored a paradox: Although most of our society classifies mathematics as the least accessible kind of knowledge, it is, paradoxically, the most accessible to children. In this chapter we shall encounter a similar paradox in the domain of science. We shall look at ways in which the thinking of children has more in common with “real science” than “school science” has with the thinking either of children or of scientists. And once more we shall note a double paradox in the way computers enter into and influence this state of affairs. The introduction of the computer can provide a way out of the paradoxes, but it usually is used in ways that exacerbate them by reinforcing the paradoxical ways of thinking about knowledge, of thinking about “school math” and “school science.”
数学水平高超的成年人会使用某些隐喻来谈论重要的学习经历。他们谈论如何了解一个想法、探索一个知识领域，以及如何对不久前似乎难以理解的细微差别产生敏感度。
Mathetically sophisticated adults use certain metaphors to talk about important learning experiences. They talk about getting to know an idea, exploring an area of knowledge, and acquiring sensitivity to distinctions that seemed ungraspably subtle just a little while ago.
我认为这些描述非常准确地适用于儿童的学习方式。但当我要求小学学生谈论学习时，他们使用的语言却截然不同，主要指他们学到的事实和掌握的技能。学校似乎为学生提供了一种特定的学习模式；我相信它不仅通过其说话方式，而且通过其实践来做到这一点。
I believe that these descriptions apply very accurately to the way children learn. But when I asked students in grade schools to talk about learning, they used a very different kind of language, referring mainly to facts they had learned and skills they had acquired. It seems very clear that school gives students a particular model of learning; I believe it does this not only through its way of talking but also through its practices.
技能和离散事实很容易以可控的剂量传授。它们也更容易衡量。而且，强制学习一项技能肯定比检查某人是否“了解”了某个想法更容易。学校强调学习技能和事实，学生将学习视为“学习那个”和“学习如何”也就不足为奇了。
Skills and the discrete facts are easy to give out in controlled doses. They are also easier to measure. And it is certainly easier to enforce the learning of a skill than it is to check whether someone has “gotten to know” an idea. It is not surprising that schools emphasize learning skills and facts and that students pick up an image of learning as “learning that” and “learning how.”
在海龟微观世界中学习，可以作为了解一个人的典范。在这种环境中学习的学生确实会发现事实、做出命题概括并学习技能。但主要的学习体验不是记忆事实或练习技能。相反，它是了解海龟，探索海龟能做什么和不能做什么。它类似于孩子的日常活动，例如做泥巴饼和测试父母权威的极限——所有这些都包含“了解”的成分。教师经常设置一些情境，声称孩子们实际上正在了解这个或那个概念，即使他们可能没有意识到这一点。然而，海龟是不同的——它允许孩子们有意识地将一种他们感到舒适和熟悉的学习方式应用到数学和物理中。而且，正如我们所指出的，这种学习方式使孩子更接近成熟的成人学习者的数学实践。各种形式的海龟（地板海龟、屏幕海龟和动态海龟）都能很好地扮演这个角色，因为它既是一个引人入胜的拟人化物体，又是一个强大的数学概念。作为数学和科学学习的典范，它与五年级学生比尔（在第 3 章中提到）描述的方法形成了鲜明的对比，比尔告诉我，他通过让大脑一片空白并反复思考来学习数学。
Working in Turtle microworlds is a model for what it is to get to know an idea the way you get to know a person. Students who work in these environments certainly do discover facts, make propositional generalizations, and learn skills. But the primary learning experience is not one of memorizing facts or of practicing skills. Rather, it is getting to know the Turtle, exploring what a Turtle can and cannot do. It is similar to the child’s everyday activities, such as making mudpies and testing the limits of parental authority—all of which have a component of “getting to know.” Teachers often set up situations in which they claim that children are actually getting to know this or that concept even though they might not realize it. Yet the Turtle is different—it allows children to be deliberate and conscious in bringing a kind of learning with which they are comfortable and familiar to bear on math and physics. And, as we have remarked, this is a kind of learning that brings the child closer to the mathetic practice of sophisticated adult learners. The Turtle in all its forms (floor Turtles, screen Turtles, and Dynaturtles) is able to play this role so well because it is both an engaging anthropomorphizable object and a powerful mathematical idea. As a model for what mathematical and scientific learning is about, it stands in sharp contrast to the methodology described by the fifth grader, Bill (mentioned in chapter 3), who told me that he learned math by making his mind a blank and saying it over and over.
对我来说，了解一个知识领域（比如，牛顿力学或黑格尔哲学）就像进入一个新的人群。有时，人们最初会被一堆令人眼花缭乱的无差别面孔所淹没。只有逐渐地，人们才会开始脱颖而出。在其他情况下，人们很幸运，很快就认识了一两个人，并与他们发展了重要的关系。这种好运可能来自于挑选“有趣”人的直觉，也可能来自于良好的介绍。同样，当人们进入一个新的知识领域时，最初会遇到一大堆新思想。好的学习者能够挑选出那些强大而友好的思想。其他技能较差的人则需要老师和朋友的帮助。但我们一定不要忘记，虽然好老师扮演着可以提供介绍的共同朋友的角色，但了解一个想法或一个人的实际工作不能由第三方来完成。每个人都必须掌握结识的技巧和做这件事的个人风格。
For me, getting to know a domain of knowledge (say, Newtonian mechanics or Hegelian philosophy) is much like coming into a new community of people. Sometimes one is initially overwhelmed by a bewildering array of undifferentiated faces. Only gradually do the individuals begin to stand out. On other occasions one is fortunate in quickly getting to know a person or two with whom an important relationship can develop. Such good luck may come from an intuitive sense for picking out the “interesting” people, or it may come from having good introductions. Similarly, when one enters a new domain of knowledge, one initially encounters a crowd of new ideas. Good learners are able to pick out those that are powerful and congenial. Others who are less skillful need help from teachers and friends. But we must not forget that while good teachers play the role of mutual friends who can provide introductions, the actual job of getting to know an idea or a person cannot be done by a third party. Everyone must acquire skill at getting to know and a personal style for doing it.
这里我们用物理学的一个例子来聚焦知识领域作为强大思想社区的形象，并在此过程中向强大思想的认识论迈进了一步。海龟微观世界说明了帮助新手开始在这样一个社区中交朋友的一些一般策略。第一个策略是确保学习者有一个这种学习的模型；使用海龟就是一个很好的例子。这一策略并不要求所有知识都“海龟化”或“简化”为计算术语。这个想法是，早期使用海龟的经验是一种很好的方式，可以通过“了解”其强大的思想来“了解”学习正式学科的感觉。我在第 3 章中提出了类似的观点，当时我建议海龟几何可能是向学习者介绍波利亚关于启发式思想的绝佳领域。这并不意味着启发式思维依赖于海龟或计算机。一旦彻底“了解”了波利亚的思想，它们就可以应用于其他领域（甚至算术）。我们在第 4 章中的讨论表明，理论物理可能是一种重要的元知识的良好载体。如果是这样，这将对我们对其在儿童生活中的作用的文化观点产生重要影响。我们可能会认为它是一门适合早期习得的学科，不仅仅是因为它解释了事物的世界，还因为它以一种让孩子更好地掌握自己的学习过程的方式解释了事物的世界。
Here we use an example from physics to focus the image of a domain of knowledge as a community of powerful ideas, and in doing so take a step toward an epistemology of powerful ideas. Turtle microworlds illustrate some general strategies for helping a newcomer begin to make friends in such a community. A first strategy is to ensure that the learner has a model for this kind of learning; working with Turtles is a good one. This strategy does not require that all knowledge be “Turtle-ized” or “reduced” to computational terms. The idea is that early experience with Turtles is a good way to “get to know” what it is like to learn a formal subject by “getting to know” its powerful ideas. I made a similar point in chapter 3 when I suggested that Turtle geometry could be an excellent domain for introducing learners to Polya’s ideas about heuristics. This does not make heuristic thinking dependent on turtles or computers. Once Polya’s ideas are thoroughly “known,” they can be applied to other domains (even arithmetic). Our discussion in chapter 4 suggested that theoretical physics may be a good carrier for an important kind of meta-knowledge. If so, this would have important consequences for our cultural view of its role in the lives of children. We might come to see it as a subject suitable for early acquistion not simply because it explicates the world of things but because it does so in a way that places children in better command of their own learning processes.
对于某些人来说，以物理学作为分析问题的模型，就等同于高度量化、形式化的方法。事实上，当心理学和社会学等学科以物理学作为模型时，结果往往不尽如人意。但所使用的物理学类型却有很大不同。对社会科学产生不良影响的物理学强调实证主义的科学哲学。我所说的是一种物理学，它使我们与实证主义观点形成鲜明对比，实证主义认为科学是一组事实和“规律”的真实断言。科学的命题内容当然非常重要，但它只是物理学家知识体系的一部分。它不是历史上最先发展起来的部分，也不是在学习过程中最先可以理解的部分，当然，它也不是我在这里提出的反思我们自身思维的模型。我们应该对更定性、更不完全指定、很少以命题形式陈述的知识感兴趣。如果给学生提供这样的方程式，如f = ma 、 E = IR或 PV = RT作为构成物理学知识的主要模型，他们处于这样一种境地：他们头脑中没有任何东西可能被识别为“物理学”。我们已经看到，这种事情使他们作为学习者面临非常高的风险。他们正在走向分离学习。他们正在把自己归类为无法理解物理学。通过与海龟一起工作，可以获得对构成物理学的知识的不同认识：在这里，一个孩子，甚至一个只拥有一块零碎的、不完全指定的定性知识的孩子（例如“这些海龟只理解变化的速度”）已经可以用它做一些事情。事实上，他或她可以开始解决困扰大学生的许多概念问题。即使不知道如何定量表示速度，也可以使用知识碎片！它是一种直觉的、非正式的，但往往非常强大的想法，无论我们是孩子还是物理学家，这些想法都存在于我们所有人的头脑中。
For some people, taking physics as a model for how to analyze problems is synonymous with a highly quantitative, formalistic approach. And indeed, the story of what has happened when such disciplines as psychology and sociology have taken physics as a model has often had unhappy endings. But there is a big difference in the kind of physics used. The physics that had a bad influence on social sciences stressed a positivistic philosophy of science. I am talking about a kind of physics that places us in firm and sharp opposition to the positivistic view of science as a set of true assertions of fact and of “law.” The propositional content of science is certainly very important, but it constitutes only a part of a physicist’s body of knowledge. It is not the part that developed first historically, it is not a part that can be understood first in the learning process, and it is, of course, not the part I am proposing here as a model for reflection about our own thinking. We shall be interested in knowledge that is more qualitative, less completely specified, and seldom stated in propositional form. If students are given such equations as f = ma, E = IR, or PV = RT as the primary models of the knowledge that constitutes physics, they are placed in a position where nothing in their own heads is likely to be recognized as “physics.” We have already seen that this is the kind of thing that puts them at very high risk as learners. They are on the road to dissociated learning. They are on the road to classifying themselves as incapable of understanding physics. A different sense of what kind of knowledge constitutes physics is obtained by working with Turtles: Here a child, even a child who possesses only one piece of fragmentary, incompletely specified, qualitative knowledge (such as “these Turtles only understand changing velocities”) can already do something with it. In fact, he or she can start to work through many of the conceptual problems that plague college students. The fragment of knowledge can be used without even knowing how to represent velocities quantitatively! It is of a kind with the intuitive and informal, but often very powerful, ideas that inhabit all of our heads whether we are children or physicists.
这种利用计算机创造定性思维机会的做法与高中物理课程中已成为标准的计算机使用方式截然不同。高中物理课程中，计算机用于通过允许进行更复杂的计算来强化物理学的定量方面。因此，它具有我们已经注意到的一些悖论，即利用新技术来强化教育方法，而这种教育方法的存在本身就反映了前计算机时代的局限性。如前所述，需要进行算术练习和实践，这是数学协同学习条件缺失的症状。计算机的正确使用就是提供这样的条件。当计算机被用来治疗算术成绩差的直接症状时，它们会强化分离学习的习惯。这些习惯延伸到生活的许多领域，是一个比算术薄弱更严重的问题。治疗可能比疾病更糟糕。物理学也有类似的论点。传统的物理教学被迫过分强调定量，这是由于纸笔技术的出现，这种技术更倾向于能够产生明确“答案”的工作。这种教学体系使用“实验室”来加强这种做法，在实验室中，人们进行实验来证明、反驳和“发现”已知的命题。这使得学生很难找到一种建设性地将直觉和形式方法结合起来的方法。每个人都忙于遵循菜谱。同样，就像算术一样，计算机应该被用来消除根本问题。然而，就目前的情况而言，学校物理作为定量的既定形象和计算机的既定形象是相互强化的。计算机的使用加剧了物理课已经过于量化的方法论。就像算术练习和实践一样，这种计算机的使用无疑会带来局部改进，因此得到了教育测试界和那些没有机会看到更好的东西的教师的认可。但在本书中，我们一直在开发一种不太量化的计算机教育方法。现在，我们直接讨论这种方向的转变必然会给严肃的物理教师带来哪些担忧。
This use of the computer to create opportunities for the exercise of qualitative thinking is very different from the use of computers that has become standard in high school physics courses. There it is used to reinforce the quantitative side of physics by allowing more complex calculations. Thus it shares some of the paradox we have already noted in the use of new technologies to reinforce educational methods whose very existence is a reflection of the limitations of the precomputer period. As previously mentioned, the need for drill and practice in arithmetic is a symptom of the absence of conditions for the syntonic learning of mathematics. The proper use of computers is to supply such conditions. When computers are used to cure the immediate symptom of poor scores in arithmetic, they reinforce habits of dissociated learning. And these habits which extend into many areas of life are a much more serious problem than weakness in arithmetic. The cure may be worse than the disease. There is an analogous argument about physics. Traditional physics teaching is forced to overemphasize the quantitative by the accidents of a paper-and-pencil technology which favors work that can produce a definite “answer.” This is reinforced by a teaching system of using “laboratories” where experiments are done to prove, disprove, and “discover” already known propositions. This makes it very difficult for the student to find a way to constructively bring together intuitions and formal methods. Everyone is too busy following the cookbook. Again, as in the case of arithmetic, the computer should be used to remove the fundamental problem. However, as things are today, the established image of school physics as quantitative and the established image of the computer reinforce each other. The computer is used to aggravate the already too-quantitative methodology of the physics classes. As in the case of arithmetic drill and practice, this use of the computer undoubtedly produces local improvements and therefore gets the stamp of approval of the educational testing community and of teachers who have not had the opportunity to see something better. But throughout this book we have been developing the elements of a less quantitative approach to computers in education. Now we directly address the concerns this shift in direction must raise for a serious teacher of physics.
本章开头的引文是一位老师在痛苦中说出的，她显然喜欢和海龟一起工作，但无法将其与她所定义的“做物理”相协调。这种情况反映了任何希望在教育领域进行彻底创新的人所面临的永久困境。创新需要新的想法。我曾说过，我们应该准备对传统知识领域进行深远的重新概念化。但这能走多远？教育对传统负有责任。例如，英语教师群体的工作必须是引导学生了解现有的和历史发展的语言和文学。如果他们发明了一种新的语言，写了他们版本的诗歌，并把这些虚构的实体代替传统实体传给下一代，那么他们就没有尽到自己的责任。这位老师担心和海龟一起工作是否是“真正学习物理”，她的担忧非常严重。
The quotation at the beginning of this chapter was spoken in some anguish by a teacher who manifestly liked working with Turtles but could not reconcile it with what she had come to define as “doing physics.” The situation reflects a permanent dilemma faced by anyone who wishes to produce radical innovation in education. Innovation needs new ideas. I have argued that we should be prepared to undertake far-reaching reconceptualizations of classical domains of knowledge. But how far can this go? Education has a responsibility to tradition. For example, the job of the community of English teachers must be to guide their students to the language and literature as it exists and as it developed historically. They would be failing in their duty if instead they invented a new language, wrote their version of poetry, and passed on to the next generation these fabricated entities in the place of the traditional ones. The concern of the teacher worried about whether working with Turtles is “really learning physics” is very serious.
学习《海龟》是否相当于用“更简单”的虚构文学作品取代莎士比亚的作品？它是否让学生接触到伽利略、牛顿和爱因斯坦的智慧成果，还是仅仅接触到一种既不伟大也未经时间考验的特殊发明？这个问题引出了一系列基本问题，其中包括：什么是物理学？计算对理解物理学的潜在影响是什么？
Is work with Turtles analogous to replacing Shakespeare by “easier,” made-up literature? Does it bring students into contact with the intellectual products of Galileo, Newton, and Einstein or merely with an idiosyncratic invention that is neither marked by greatness nor tested by time? The question raises fundamental problems, among them: What is physics? And what is the potential influence of computation on understanding it?
大多数课程设计者对这些问题都有简单的答案。他们将基础物理定义为学校教授的内容。有时，他们会将通常在大学教授的内容移到高中，或引入与旧内容相同的新主题。例如，教科书提到了现代粒子，并简要介绍了核反应堆的工作原理。即使是更有远见的课程改革者也停留在由方程式、定量定律和实验室实验定义的概念框架内。因此，他们可以放心，他们真的在“教物理”。计算机开启了一种新的活动和一种与思想的新关系的可能性，这带来了对文化遗产的责任问题。我认真对待这一责任，但不能觉得我通过躲在现有课程后面来履行这一责任。如果不认真考虑这样一个问题，就不能接受这种庇护：学校科学是否已经处于假设的英语教师的境地，他们教授一种代用英语，因为它似乎更容易教。我相信情况确实如此。
Most curriculum designers have easy answers to these questions. They define elementary physics as what is taught in schools. Occasionally they move material usually taught in college down to high school, or bring in new topics of the same kind as the old. For example, modern particles are mentioned and the textbooks show schematically how a nuclear reactor works. Even the more visionary curriculum reformers stayed within the conceptual framework defined by equations, quantitative laws, and laboratory experiments. Thus, they could feel secure that they were really “teaching physics.” The possibility opened by the computer of a new kind of activity and of a new relationship to ideas poses problems of responsibility toward the cultural heritage. I take this responsibility seriously but cannot feel that I serve it by taking shelter behind the existing curriculum. One cannot accept this shelter without seriously considering the question of whether school science is not already in the position of the hypothetical English teacher who taught an ersatz form of English because it seemed to be more teachable. I believe that this is the case.
在第 5 章中，我指出，背叛“真实物理”精神的是“学校物理”，而不是“海龟物理”。在这里，我通过讨论比 Dynaturtles 更远离传统课程的物理学组成部分来进一步论证我的观点。这些都是非常笼统的、通常是定性的、直观的想法或“框架”，物理学家在决定应用哪些定量原理之前会用它们来思考问题。
In chapter 5, I suggested that it is “school physics” rather than “Turtle physics” that betrays the spirit of “real physics.” Here I pursue my argument by talking about components of physics that are even further removed than Dynaturtles from the traditional curriculum. These are very general, usually qualitative, intuitive ideas or “frames” used by physicists to think about problems before they can even decide what quantitative principles apply.
我请那些可能不熟悉物理学中这种定性思维的读者来听听两位伟大物理学家之间的假设对话。
I ask readers who may not be familiar with such qualitative thinking in physics to follow a hypothetical conversation between two great physicists.
数以百万计的学生从小就相信伽利略通过从比萨斜塔上扔下炮弹，驳斥了亚里士多德的期望，即物体落地所需的时间与其重量成正比。伽利略的实验据称证明，除了由于空气阻力而产生的微小扰动外，一颗重的炮弹和一颗轻的炮弹如果一起落地，会同时落地。事实上，伽利略极不可能做过这样的实验。但他是否做过并不那么有趣，重要的是他对实验结果没有丝毫怀疑。为了传达一种让他有这种信心的思维方式，我们将讨论两个虚构人物GAL和ARI之间的假设对话。
Many millions of students have grown up believing that Galileo refuted Aristotle’s expectation that the time taken for an object to fall to the ground is proportional to its weight by dropping cannonballs from the tower of Pisa. Galileo’s experiment is supposed to have proved that except for minor perturbations due to air resistance, a heavy and a light cannonball would, if dropped together, reach the ground together. In fact it is extremely unlikely that Galileo performed any such experiment. But whether he did or did not is less interesting than the fact that he would not have had the slightest doubt about the outcome of the experiment. In order to convey a sense of the kind of thinking that could have given him this assurance, we shall go through a hypothetical dialog between two imaginary characters, GAL and ARI.
加尔各答： GAL: |
瞧，你的理论肯定是错的。这里有一个两磅重的球和一个一磅重的球。两磅重的球需要两秒钟才能落地。告诉我，你认为一磅重的球需要多长时间？ Look, your theory has got to be wrong. Here’s a two-pound and a one-pound ball. The two-pound ball takes two seconds to fall to the ground. Tell me, how long do you think the one-pound ball would need? |
阿里： ARI: |
我估计需要四秒钟。无论如何，远不止两秒钟。 I suppose it would take four seconds. Anyway, much more than two seconds. |
加尔各答： GAL: |
我以为你会这么说。但现在请回答另一个问题。我即将同时投掷两枚一磅炸弹。它们需要多长时间才能落地？ I thought you would say that. But now please answer another question. I am about to drop two one-pounders simultaneously. How long will the pair of them take to reach the ground? |
阿里： ARI: |
这不是另一个问题。我给出了我的观点，一磅重的球需要四秒钟。其中两个必须用同样的时间。每个球都独立落下。 That’s not another question. I gave my opinion that one-pound balls take four seconds. Two of them must do the same. Each falls independently. |
加尔各答： GAL: |
如果两个身体是两个身体而不是一个身体，那么您就与自己一致。 You are consistent with yourself if two bodies are two bodies, not one. |
阿里： ARI: |
就像他们一样......当然。 As they are… of course. |
加尔各答： GAL: |
但现在如果我用一条细线把它们连起来……这是两具尸体还是一具尸体？它（或它们）落地需要两秒还是四秒？ But now if I connect them by a gossamer thread… is this now two bodies or one? Will it (or they) take two seconds or four to fall to the ground? |
阿里： ARI: |
我真的很困惑。让我想想……它是一个物体，但它应该在落地前下落四秒钟。但这意味着比丝绸更细的线可以减缓一个猛烈下落的铁球的速度。这似乎是不可能的。但如果我说它是两个物体……我就很困惑了。什么是物体？我怎么知道什么时候一个物体变成两个？如果我不知道，那么我怎么能确定我的下落物体定律呢？ I am truly confused. Let me think.… It’s one body, but then it should fall for four seconds before reaching the earth. But then this would mean that a thread finer than silk could slow down a furiously falling ball of iron. It seems impossible. But if I say it is two bodies… I am in deep trouble. What is a body? How do I know when one becomes two? And if I cannot know, then how sure can I be of my laws of falling bodies? |
从严格的逻辑角度来看，GAL的论证并非绝对令人信服。我们可以想象ARI理论的“修正”。例如，他可以提出所花的时间可能取决于物体的形状和重量。这将使他得出一个结论，即由两颗炮弹和游丝组成的两磅重的物体下落的速度会比两磅重的铁球慢。但事实上GAL使用的论证颠覆了ARI阐述的理论，而且从历史上看，这种论证极有可能推动了对亚里士多德思想的伟大转变。没有任何一个论证能够单独改变亚里士多德的思想，对亚里士多德来说，物体下落理论是一个相互支持的网络中的一个元素。但随着GAL的思维方式流行起来，亚里士多德体系受到了侵蚀。事实上，我认为，与来自精确事实和方程式的显然更有说服力的论据相反，这种论据在思维的演变中发挥着至关重要的作用，无论是在科学本身演变的历史尺度上，还是在个体学习者发展的个人尺度上。
From a strictly logical point of view, GAL’s argument is not absolutely compelling. One can imagine “fixes” for ARI’s theory. For example, he could propose that the time taken might depend on the form as well as the weight of the body. This would allow him the possibility that a two-pound body made of two cannonballs and gossamer threads would fall more slowly than a two-pound sphere of iron. But in fact the kind of argument used by GAL is subversive of the kind of theory expounded by ARI, and historically, it is highly plausible that the great conversion from Aristotelian thinking was fueled by such arguments. No single argument could by itself convert Aristotle, for whom the theory of falling objects was an element in a mutually supporting web. But as GAL’s way of thinking gained currency, the Aristotelian system was eroded. Indeed I contend that arguments of this kind, as opposed to the apparently more compelling arguments from precise facts and equations, play an essential role in the evolution of thinking, both on the historical scale of the evolution of science itself and on the personal scale of the development of the individual learner.
如果GAL论证的是具体事实或计算， ARI就能够更好地为自己辩护，因为这可能会引起对其适用条件的争论，并允许它们被区分开来。GAL 论证的强有力之处在于它调动了ARI自己对物理对象的性质和自然效应的连续性的直觉（比丝绸更薄，而不是猛烈下落的铁）。对于逻辑学家来说，这个论点可能看起来不那么令人信服。但作为有同情心的人类同胞，我们发现自己对ARI感到困惑不已。
ARI would have been far better able to defend himself had GAL argued from specific facts or calculations, which might allow quibbles about their conditions of applicability and allow themselves to be compartmentalized. The hard punch of GAL’s argument comes from the fact that it mobilizes ARI’s own intuitions about the nature of physical objects and about the continuity of natural effects (thinner than silk versus furiously falling iron). To a logician this argument might seem less compelling. But as empathetic fellow humans we find ourselves squirming in confusion with ARI.
通过思考这个对话提出的问题，我们可以学到很多东西，尽管它过于简单。首先，我们注意到GAL不仅比ARI聪明：他知道一些ARI似乎不知道的东西。事实上，如果我们仔细观察，就会发现GAL巧妙地运用了几个强有力的想法。最引人注目的是他的主要思想，即把一个两磅重的物体看作是由两个一磅重的物体组成的，认为整体是由我们想把它分成的任何部分相加而成的。抽象地说，这个想法在某些情况下听起来微不足道，在其他情况下则完全是错误的：我们习惯于被提醒“整体大于部分之和”。但我们不应该把它当作一个要用真假标准来判断的命题。它是一个想法，一个智力工具，一个已被证明在巧妙使用时非常强大的工具。
There is a lot to be learned by thinking through the issues raised by this dialogue, simplistic as it is. First we note that GAL is not just being cleverer than ARI: He knows something that ARI seems not to know. In fact, if we look carefully we see that GAL skillfully deploys several powerful ideas. Most striking is his principal idea of looking at a two-pound object as made up of two one-pound objects, seeing the whole as additively made of whatever parts we care to divide it into. Stated abstractly this idea sounds trivial in some contexts and simply false in others: We are used to being reminded that “the whole is more than the sum of its parts.” But we should not treat it as a proposition to be judged by the criterion of truth and falsity. It is an idea, an intellectual tool, and one that has proved itself to be enormously powerful when skillfully used.
GAL的思想非常强大，是每一位现代数学家、物理学家或工程师的智力工具包的一部分。它在历史和物理学习中的重要性不亚于适合命题或方程式的知识。但人们无法从教科书中知道这一点。GAL的思想没有名字，没有归因于历史科学家，教师们对它只字未提。事实上，与大多数直觉物理学一样，这种知识似乎是成年物理学家通过皮亚杰学习过程获得的，没有经过刻意的课堂教学，而且往往与课堂教学背道而驰。当然，我之所以承认这些非正式学习的、强大的直觉思想的存在，并不是要将它们从皮亚杰学习的范围中移除，并将它们放入课程中：还有其他方法可以促进它们的获得。通过承认它们的存在，我们应该能够创造条件促进它们的发展，我们当然可以做很多事情来消除许多传统学习环境中阻碍它们的障碍。
GAL’s idea is powerful and is part of the intellectual tool kit of every modern mathematician, physicist, or engineer. It is as important in the history and in the learning of physics as the kind of knowledge that fits into propositions or equations. But one would not know this from looking at textbooks. GAL’s idea is not given a name, it is not attributed to a historical scientist, it is passed over in silence by teachers. Indeed, like most of intuitive physics, this knowledge seems to be acquired by adult physicists through a process of Piagetian learning, without, and often in spite of, deliberate classroom teaching. Of course, my interest in recognizing the existence of these informally learned, powerful intuitive ideas is not to remove them from the scope of Piagetian learning and place them in a curriculum: There are other ways to facilitate their acquisition. By recognizing their existence we should be able to create conditions that will foster their development, and we certainly can do a lot to remove obstacles that block them in many traditional learning environments.
GAL与ARI的对话让我们了解到学习中最具破坏性的障碍之一：用形式推理来压制直觉。
GAL’s dialogue with ARI has something to teach us about one of the most destructive blocks to learning: the use of formal reasoning to put down intuitions.
每个人都知道，当我们遇到违反直觉的现象时，我们被迫通过观察或推理承认现实并不符合我们的预期，这种感觉会让人不快。许多人在面对牛顿粒子的永恒运动、船舵转动船只的方式或玩具陀螺的奇怪行为时，都会有这种感觉。在所有这些情况下，直觉似乎背叛了我们。有时有一个简单的“解决办法”；我们发现自己犯了一个表面上的错误。但有趣的是，无论我们对问题进行多少思考，冲突仍然顽固地存在。在这些情况下，我们很容易得出“直觉不可信任”的结论。在这些情况下，我们需要改进我们的直觉，对其进行调试，但我们面临的压力是放弃直觉，转而依赖方程式。通常，当一个处于这种困境的学生去找物理老师说：“我认为陀螺仪应该倒下而不是直立”，老师会写一个方程来证明陀螺仪是直立的。但这不是学生需要的。他已经知道陀螺仪会直立，而这种知识与直觉相冲突，让他很受伤。通过证明陀螺仪会直立，老师在伤口上撒盐，但并没有采取任何措施来治愈伤口。学生需要的是完全不同的东西：更好地了解自己，而不是陀螺仪。他想知道为什么他的直觉给了他错误的期望。他需要知道如何利用直觉来改变它们。从对话中我们可以看出，GAL是操纵直觉的专家。他并没有强迫ARI拒绝直觉而选择计算。相反，他迫使他面对直觉思维的一个非常具体的方面：他如何思考物体。从对话中可以推测，GAL习惯于将对象视为由各个部分或子对象组成，而ARI则习惯于更全面地思考对象，将对象视为具有形状和重量等整体属性的不可分割的整体。
Everyone knows the unpleasant feeling evoked by running into a counterintuitive phenomenon where we are forced, by observation or by reason, to acknowledge that reality does not fit our expectations. Many people have this feeling when faced with the perpetual motion of a Newtonian particle, with the way a rudder turns a boat, or with the strange behavior of a toy gyroscope. In all these cases intuition seems to betray us. Sometimes there is a simple “fix”; we see that we made a superficial mistake. But the interesting cases are those where the conflict remains obstinately in place however much we ponder the problem. These are the cases where we are tempted to conclude that “intuition cannot be trusted.” In these situations we need to improve our intuition, to debug it, but the pressure on us is to abandon intuition and rely on equations instead. Usually when a student in this plight goes to the physics teacher saying, “I think the gyroscope should fall instead of standing upright,” the teacher responds by writing an equation to prove that the thing stands upright. But that is not what the student needed. He already knew that it would stay upright, and this knowledge hurt by conflicting with intuition. By proving that it will stand upright the teacher rubs salt in the wound but does nothing to heal it. What the student needs is something quite different: better understanding of himself, not of the gyroscope. He wants to know why his intuition gave him a wrong expectation. He needs to know how to work on his intuitions in order to change them. We see from the dialogue that GAL is an expert at how to manipulate intuitions. He does not force ARI into rejecting intuition in favor of calculation. Rather he forces him to confront a very specific aspect of his intuitive thinking: how he thinks about objects. One suspects from the dialogue that GAL is used to understanding objects by thinking of them as composed of parts, or subobjects, while ARI is used to thinking of objects more globally, as undivided wholes with global properties such as shape and weight.
我们似乎已经偏离了对计算机的讨论。但GAL和ARI之间的互动接近于儿童与计算机之间以及儿童与教师之间通过计算机进行的一种重要互动。GAL试图让ARI面对并运用他对物体的直觉思维方式，而ARI可能足够熟练地做到这一点。但孩子们能做些什么来面对他们的直觉呢？
We might seem to have strayed far from our discussion of computers. But the interaction between GAL and ARI is close to an important kind of interaction between children and computers and between children and instructors via computers. GAL tried to make ARI confront and work through his intuitive ways of thinking about objects, and ARI might be skillful enough to do so. But what can children do to confront their intuitions?
当然，这个问题是反问的，因为我知道孩子们会花很多心思思考自己的思维。他们确实会担心自己的直觉。他们确实会面对直觉，也确实会调试直觉。如果他们不这样做，那么让他们这样做的想法确实是不切实际的。但既然他们已经这样做了，我们可以提供材料来帮助他们做得更好。
Of course the question is rhetorical in that I know that children think a great deal about their thinking. They do worry about their intuitions. They do confront them and they do debug them. If they did not, the idea of making them do so would indeed be utopian. But since they do it already, we can provide materials to help them do it better.
我认为计算机在两个方面有所帮助。首先，计算机允许或迫使孩子将直觉期望外化。当直觉被转化为程序时，它变得更加引人注目，也更容易被反思。其次，计算思想可以作为重塑直觉知识的材料。下面对一个著名谜题的分析用于说明海龟模型如何帮助弥合形式知识和直觉理解之间的差距。我们在儿童使用计算机的事件中看到了很多例子。在这里，我将通过邀请您处理您的直觉会发生冲突的情况来传达这意味着什么。
I see the computer as helping in two ways. First, the computer allows, or obliges, the child to externalize intuitive expectations. When the intuition is translated into a program it becomes more obtrusive and more accessible to reflection. Second, computational ideas can be taken up as materials for the work of remodeling intuitive knowledge. The following analysis of a well-known puzzle is used to illustrate how a Turtle model can help bridge the gap between formal knowledge and intuitive understanding. We have seen many examples in incidents where children work with computers. Here I shall convey a sense of what this means by inviting you to work on a situation where your intuitions will come into conflict.
解决问题的目的不是“得到正确的答案”，而是敏锐地寻找不同思考方式之间的冲突：例如，两种直觉思考方式之间或直觉分析与形式分析之间的冲突。当你意识到冲突时，下一步就是解决它们，直到你感觉更舒服为止。当我这样做时，我发现海龟模型在解决一些冲突方面非常有帮助。但我的反应无疑是由我对海龟的积极感受所决定的。
The purpose in working on the problem is not to “get the right answer,” but to look sensitively for conflict between different ways of thinking about the problem: for example, between two intuitive ways of thinking or between an intuitive and a formal analysis. When you recognize conflicts, the next step is to work through them until you feel more comfortable. When I did this, I found that the Turtle model was extremely helpful in resolving some of the conflicts. But my reaction is undoubtedly shaped by my positive feelings about Turtles.
想象一下，有一根绳子绕着地球的圆周，为此，我们将地球视为一个半径为 4000 英里的完全光滑的球体。有人提议将绳子放在六英尺高的杆子上。显然，这意味着绳子必须更长。关于绳子必须长多少，人们展开了讨论。大多数上过高中的人都知道如何计算答案。但在这样做或继续阅读之前，请试着猜一下：它长约 1000 英里、约 100 英里还是约 10 英里？
Imagine a string around the circumference of the earth, which for this purpose we shall consider to be a perfectly smooth sphere, four thousand miles in radius. Someone makes a proposal to place the string on six-foot-high poles. Obviously this implies that the string will have to be longer. A discussion arises about how much longer it would have to be. Most people who have been through high school know how to calculate the answer. But before doing so or reading on, try to guess: Is it about one thousand miles longer, about a hundred, or about ten?
大多数有先思考后计算的习惯的人（这种习惯是调试直觉的诀窍的一部分）都会有一种强烈的直觉，认为需要“很多”额外的字符串。对于某些人来说，这种信念的来源似乎在于地球周长大约两万四千英里周围正在添加一些东西。其他人则将其与更抽象的比例考虑联系起来。但无论这种信念的来源是什么，它在预测正式计算的结果时都是“错误的”，结果发现结果不到四十英尺。直觉和计算之间的冲突如此强烈，以至于这个问题被广泛称为难题。从这种冲突中得出的结论往往是直觉不可信。我们不会得出这个结论，而是尝试与读者进行对话，以确定需要做些什么来改变这种直觉。
Most people who have the discipline to think before calculating—a discipline that forms part of the know-how of debugging one’s intuitions—experience a compelling intuitive sense that “a lot” of extra string is needed. For some the source of this conviction seems to lie in the idea that something is being added all around the twenty-four thousand miles (or so) of the earth’s circumference. Others attach it to more abstract considerations of proportionality. But whatever the source of the conviction may be, it is “incorrect” in anticipating the result of the formal calculation, which turns out to be a little less than forty feet. The conflict between intuition and calculation is so powerful that the problem has become widely known as a teaser. And the conclusion that is often drawn from this conflict is that intuitions are not to be trusted. Instead of drawing this conclusion, we shall attempt to engage the reader in a dialog in order to identify what needs to be done to alter this intuition.
图 15
Figure 15
图中显示一根绳子绕着地球，由高度夸张的杆子支撑。地球半径为 R，杆子高度为 h。问题是估算外圆周长和真实圆周长之间的长度差。这很容易通过以下公式计算出来：
The figure shows a string around the earth supported by poles of greatly exaggerated height. Call the radius of the earth R and the height of the poles h. The problem is to estimate the difference in length between the outer circumference and the true circumference. This is easy to calculate from the formula:
周长 = 2π × 半径
CIRCUMFERENCE = 2π × RADIUS
所以差异一定是
So the difference must be
2π(R + h) − 2πR
2π(R + h) − 2πR
也就是2πh。
which is simply 2πh.
但这里的挑战是“直觉”出一个近似的答案，而不是“计算”出一个确切的答案。
But the challenge here is to “intuit” an approximate answer rather than to “calculate” an exact one.
首先，我们遵循寻找可能更易处理的类似问题的原则。简化的一个好通用规则是寻找线性版本。因此，我们假设“地球是方形的”，提出同样的问题。
As a first step we follow the principle of seeking out a similar problem that might be more tractable. And a good general rule for simplification is to look for a linear version. Thus we pose the same problem on the assumption of a “square earth.”
图 16a
Figure 16a
假设杆子上的绳子与正方形的距离为 h 。绳子沿着边是直的。当它绕过角落时，它会沿着半径为h的圆周运动。绳子的直线部分与正方形的边长相同。多余的长度都在角落，在四个四分之一圆的饼状切片中。四个四分之一圆组成一个半径为h的整个圆。所以“多余的绳子”就是这个圆的周长，也就是2π h。
The string on poles is assumed to be at distance h from the square. Along the edges the string is straight. As it goes around the corner it follows a circle of radius h. The straight segments of the string have the same length as the edges of the square. The extra length is all at the corners, in the four quarter-circle pie slices. The four quarter circles make a whole circle of radius h. So the “extra string” is the circumference of this circle, that is to say 2πh.
图 16b
Figure 16b
增加正方形的尺寸不会改变四分之一圆饼片。因此，对于非常小的正方形地球和非常大的正方形地球，将绳子从地面提升到高度h所需的额外绳子是相同的。
Increasing the size of the square does not change the quarter-circle pie slices. So the extra string needed to raise a string from the ground to height h is the same for a very small square earth as for a very large one.
该图为我们提供了一种几何学方法，让我们看到这里需要的额外绳子的数量与圆的情况相同。这本身就令人吃惊。但更令人吃惊的是，我们可以如此直接地看到，正方形的大小对需要多少额外绳子没有影响。我们可以通过公式计算出这个事实。但这样做会让我们陷入同样的困境。通过几何学的“观察”，我们可以将这种情况与我们的直观原则相一致：只有在地球弯曲的地方才需要额外的绳子。显然，将直线从地面升到六英尺高不需要额外的绳子。
The diagram gives us a geometric way to see that the same amount of extra string is needed here as in the case of the circle. This is itself quite startling. But more startling is the fact that we can see so directly that the size of the square makes no difference to how much extra string is needed. We could have calculated this fact by formula. But doing so would have left us in the same difficulty. By “seeing” it geometrically we can bring this case into line with our intuitive principle: Extra string is needed only where the earth curves. Obviously no extra string is needed to raise a straight line from the ground to a six-foot height.
不幸的是，这种理解正方形的方式似乎会破坏我们对圆形的理解。我们完全理解了正方形，但这样做是因为我们把它看作与圆形有很大的不同。
Unfortunately, this way of understanding the square case might seem to undermine our understanding of the circular case. We have completely understood the square but did so by seeing it as being very much different from the circle.
但是还有另一个强有力的想法可以解决这个问题。这就是中间情况的想法：当两种情况之间存在冲突时，寻找中间情况，正如GAL在两个一磅重的球和一个两磅重的球之间构建一系列中间对象时所做的那样。但是正方形和圆形之间的中间是什么？任何学过微积分或海龟几何的人都会立即给出答案：边越来越多的多边形。因此，我们看一下图 17，其中显示了围绕一系列多边形地球的弦。我们看到，在所有这些情况下，所需的额外弦保持不变，而且，值得注意的是，我们看到了一些可能会削弱圆形在各处都增加一些东西的论点的东西。1000 边形在比正方形多得多的地方增加了一些东西，事实上是正方形的二百五十倍。但它增加的东西较少，事实上在每个地方都只有二百五十分之一。
But there is another powerful idea that can come to the rescue. This is the idea of intermediate cases: When there is a conflict between two cases, look for intermediates, as GAL in fact did in constructing a series of intermediate objects between the two one-pound balls and one two-pound ball. But what is intermediate between a square and a circle? Anyone who has studied calculus or Turtle geometry will have an immediate answer: polygons with more and more sides. So we look at Figure 17, which shows strings around a series of polygonal earths. We see that the extra string needed remains the same in all these cases, and, remarkably, we see something that might erode the argument that the circle adds something all around. The 1000-gon adds something at many more places than the square, in fact two hundred fifty times as many places. But it adds less, in fact one two hundred fiftieth at each of them.
现在你的思维会跳动起来吗？像GAL一样，到目前为止，我还没有说过要用严密的逻辑来强制完成这一关键步骤。我也不会。但到了这个时候，有些人开始动摇了，我猜想他们是否动摇取决于他们对多边形近似圆的理念的坚定程度。对于那些已经掌握了多边形表示的人来说，多边形和圆的等价性是如此直接，以至于直觉会随之而来。那些还没有“掌握”多边形表示和圆之间等价性的人可以努力更好地熟悉它，例如，通过用它来思考其他问题。
Now will your mind take the jump? Like GAL, I have said nothing so far to compel this crucial step by rigorous logic. Nor shall I. But at this point some people begin to waver, and I conjecture that whether they do or not depends on how firm a commitment they have made to the idea of polygonal approximations to a circle. For those who have made the polygonal representation their own, the equivalence of polygon and circle is so immediate that intuition is carried along with it. People who do not yet “own” the equivalence between polygonal representation and circle can work at becoming better acquainted with it, for example, by using it to think through other problems.
图 17
Figure 17
在八边形中，“额外的线”也全部在角落的饼状切片中。如果将它们放在一起，它们会形成一个半径为h的圆。与正方形的情况一样，无论八边形是小还是大，这个圆都是一样的。适用于正方形（4 边形n）和八边形（8 边形n）的方法也适用于100 边形和1000 边形。
In the octagon, too, the “extra string” is all in the pie slices at the corners. If you put them together they form a circle of radius h. As in the case of the square, this circle is the same whether the octagon is small or big. What works for the square (4-gon) and for the octagon (8-gon) works for the 100-gon and for the 1000-gon.
以下问题摘自马丁·加德纳 (Martin Gardner) 的书《数学嘉年华》：
The following problem is taken from Martin Gardner’s book, Mathematical Carnival:
如果一枚硬币绕着另一枚硬币滚动而不滑动，那么它在一圈内会旋转多少圈？有人可能会猜测答案是一圈，因为移动的硬币沿着等于其自身周长的边缘滚动，但一个快速实验表明答案是两圈；显然，移动硬币的完整旋转会增加一次额外的旋转。1
If one penny rolls around another penny without slipping how many times will it rotate in making one revolution? One might guess the answer to be one, since the moving penny rolls along an edge equal to its own circumference, but a quick experiment shows that the answer is two; apparently the complete revolution of the moving penny adds an extra rotation.1
直觉猜测（一次旋转）和更仔细的调查结果之间再次出现冲突。如何才能使直觉保持一致？
Again there is a conflict between the intuitive guess (one revolution) and the result of more careful investigation. How can one bring one’s intuition into line?
此处的策略与绕地球绳子问题相同。将一枚硬币绕正方形滚动而不滑落。您会注意到，硬币沿边滚动时的行为与绕角旋转时的行为截然不同。很容易看出，四个角的总旋转角度加起来是 360°。对于任何多边形，无论它有多少条边，无论它有多大，这都是正确的。同样，关键的一步是从多边形到海龟圆再到真正的圆的转变。
The same strategy works here as for the string around the earth problem. Roll a penny around a square without slipping. You will notice that it behaves quite differently as it rolls along the sides than when it pivots around the corners. It is easy to see that the total rotation at the four corners combined is 360°. This remains true for any polygon, however many sides it has and however big it is. And once more, the crucial step becomes the passage from the polygon to a Turtle circle to a true circle.
我并不是说再做一次练习就能改变你对循环的直觉。在这里，就像亚里士多德物理学的情况一样，特定的知识是相互支持的思维方式的庞大网络的一部分。我建议你把这种新的思维方式记在心里一段时间，寻找机会使用它，就像你寻找机会把新朋友介绍给老朋友一样。即便如此，我也无法知道你是否想改变你对循环的直觉。但如果要改变，我认为我在这里建议的过程是最好的，也许是唯一的，不管是有意采用还是无意识地发生。
I am not suggesting that one more exercise will change your intuition of circularity. Here too, as in the case of Aristotle’s physics, the particular piece of knowledge is part of a large network of mutually supportive ways of thinking. I am suggesting that you keep this new way of thinking in mind for awhile, looking for opportunities to use it as you might look for opportunities to introduce a new friend to old ones. And even then, I have no way of knowing whether you want to change your intuition of circularity. But if it is to change, I think that the process I am suggesting here is the best, perhaps the only, way whether it is adopted deliberately or simply happens unconsciously.
我希望你们读完这本书后，能够对儿童作为思想者的价值有新的认识，甚至能够对“认识论者”的价值有新的认识，认识到强大思想的力量。但我也意识到，这些图像对你们中的某些人来说可能看起来很抽象，甚至令人恼火，也许尤其是那些教孩子的老师。
I want you to go away from this book with a new sense of a child’s value as a thinker, even as an “epistemologist” with a notion of the power of powerful ideas. But I also realize that these images might seem abstract and even irritating to some of you, perhaps especially those of you who teach children.
例如，一位三年级老师每天要花很多令人沮丧的时间教 36 个孩子写符合语法的句子和做算术，他可能会认为我关于海龟几何、物理微观世界和控制论的建议与现实相去甚远，就像玛丽·安托瓦内特建议那些饿着肚子的人应该吃蛋糕一样。我们讨论过的这些强有力的想法与大多数学校认为的谋生手段，也就是基本技能，有什么关系呢？
For example, a third-grade teacher who spends many frustrating hours every day trying to teach thirty-six children to write grammatical sentences and to do arithmetic might view my suggestions about Turtle geometry, physics microworlds, and cybernetics as far removed from reality, as far removed as was Marie Antoinette when she suggested that those who were starving for bread should eat cake. How are the powerful ideas we have discussed related to what most schools see as their bread-and-butter work, that is to say, the basic skills?
最初的联系取决于学习者的态度。如果你带着恐惧和憎恨的心理去学习，你就学不到谋生技能。当孩子不愿意让数字进入他们的大脑而无法学习算术时，补救措施必须是与数字建立一种新的关系。实现这一点可以让孩子与他们认为是同类的任何其他事物建立积极的关系。这可以是学校数学。
A first connection works through the attitude of the learner. You can’t learn bread-and-butter skills if you come to them with fear and the anticipation of hating them. When children who will not let a number into their head fail to learn arithmetic, the remedy must be developing a new relationship with numbers. Achieving this can put children in a positive relationship to anything else that they will recognize as being of the same kind. This can be school mathematics.
金是一个五年级的女孩，在学校的所有算术测试中成绩总是垫底。她讨厌数学。在LOGO环境中，她全神贯注于编程。她设计了一个项目，该项目维护一个特殊的数据库来存储有关她的家谱的信息。有一天，一位来访的教育工作者对她说：“计算机让数学变得有趣。”金放下手中的工作，非常生气地说：“数学一点也不好玩。”她班上的老师认为不宜与她讨论她用电脑做的事情是否是“数学”。显然，任何好的东西在定义上都不是数学。但到了年底，金自己找到了联系，并认定数学既不令人讨厌也不困难。
Kim was a fifth-grade girl who invariably came out on the bottom on all school arithmetic tests. She hated math. In a LOGO environment she became engrossed in programming. She designed a project that maintained a special database to store information about her family tree. One day a visiting educator remarked to her that “computers made math fun.” Kim looked up from her work and said very angrily: “There ain’t nothin’ fun in math.” The instructor in her class had not thought it advisable to discuss with her whether what she was doing with the computer was “math.” Clearly, anything that was good was definitionally not math. But by the end of the year Kim made the connection herself and decided that mathematics was neither unpleasant nor difficult.
就像你了解（并喜欢）一个人一样去了解（并喜欢）数学，这与本例非常相似。计算机还可以通过改变我们对算术的认识以及对算术中最重要的思想的认识，为基本算术的学习做出贡献。学校算术通常被认为是数论的一个分支，但它更应该被视为计算机科学的一个分支。孩子们遇到的困难通常不是因为他们对数字的概念不足，而是因为未能掌握相关的算法。学习算法可以看作是制作、使用和修复程序的过程。当一个人将多位数字相加时，他实际上是在充当计算机，执行类似图 18中程序的过程。
Getting to know (and like) mathematics as you get to know (and like) a person is a very pertinent image of what happened in this case. Computers can also contribute to the learning of bread-and-butter arithmetic by changing our perception of what it is about, of what powerful ideas are most important in it. School arithmetic, generally thought of as a branch of number theory, might better be thought of as a branch of computer science. Difficulties experienced by children are not usually due to deficiencies in their notion of number but in failing to appropriate the relevant algorithms. Learning algorithms can be seen as a process of making, using, and fixing programs. When one adds multidigit numbers, one is in fact acting as a computer in carrying through a procedure something like the program in Figure 18.
1. 按照常规格式列出数字。
1. Set out numbers following conventional format.
2. 将注意力集中在最右边的列上。
2. Focus attention on the rightmost column.
3. 对于个位数添加 as。
3. Add as for single digit numbers.
4. 如果结果<10，则记录结果。
4. If result <10 record results.
5. 如果最右边一列的结果等于或大于 10，则记录最右边的数字并在左边的下一列输入其余数字。
5. If result in rightmost column was equal to or greater than 10, then record rightmost digit and enter rest in next column to left.
6. 将注意力集中到左边一列。
6. Focus attention one column to left.
7. 转到第 3 行。
7. Go to line 3.
图 18
Figure 18
为了更好地完成这类活动，人们需要更多地了解并更适应程序的方式。当然，这正是良好的计算机经验所允许的。
To get better at this sort of activity one needs to know more about, and feel more comfortable with, the ways of procedures. And this, of course, is what a good computer experience allows.
这些言论应该放在我们之前讨论的背景下，即 20 世纪 60 年代的新数学课程改革与计算机文化能给数学带来的丰富程度之间的差异。在第二章中，我们讨论了新数学失败的一个重要原因：它没有改善我们社会与数字的疏远关系。相反，它加剧了这种关系。现在我们看到了新数学失败的第二个原因。它试图将数学教学根植于数论、集合论或逻辑，而不是面对儿童真正遇到的概念障碍：他们缺乏编程知识。因此，新数学的作者误解了儿童问题的根源。这种误解在几个方面都是有害的。它是有害的，因为它试图通过练习不相关的知识领域来提高儿童对算术的理解。它也是有害的，因为它将不适当的价值体系灌输到数学教育中。纯数学家认为数字的概念很有价值、很强大而且很重要。而程序的细节则显得肤浅和无趣。因此，孩子的困难被归结为抽象的数字概念困难。计算机科学家采取更直接的方法。加法困难并不被视为其他问题的征兆；而是加法程序的问题。对于计算机专家来说，程序及其出错的方式与其他任何事情一样有趣和概念化。此外，出错的地方，即错误，并不被视为像瘟疫一样需要避免的错误，而是学习过程的一个固有部分。
These remarks should be put in the context of our earlier discussion about the difference between the New Math curriculum reform of the 1960s and the kind of enrichment the computer culture can bring to mathematics. In chapter 2 we dealt with one important reason for the failure of the New Math: It did not ameliorate our society’s alienated relationship with number. On the contrary, it aggravated it. We now see a second reason for the failure of the New Math. It tried to root the teaching of math in number theory, set theory, or logic instead of facing the conceptual stumbling blocks that children really experience: Their lack of knowledge about programming. Thus the authors of the New Math misunderstood the source of children’s problems. This misunderstanding is harmful in several ways. It is harmful insofar as it seeks to improve the child’s understanding of arithmetic by drill in irrelevant areas of knowledge. It is also harmful insofar as it imparts an inappropriate value system into mathematics education. The pure mathematician sees the idea of number as valuable, powerful, and important. The details of procedure are seen as superficial and uninteresting. Thus the child’s difficulties are referred back to abstract difficulties with the notion of number. The computer scientist takes a more direct approach. Trouble with adding is not seen as symptomatic of something else; it is trouble with the procedure of adding. For the computerist the procedure and the ways it can go wrong are fully as interesting and as conceptual as anything else. Moreover, what went wrong, namely the bugs, are not seen as mistakes to be avoided like the plague, but as an intrinsic part of the learning process.
肯是一名五年级学生，他将 35 和 35 相加，结果为 610。他的错误显而易见。由于 32 加 32 等于 64，因此 35 加 35 应该等于 610。当肯学会将自己的错误视为数学形式主义对我们开的一个玩笑时，他与数学的关系变得更好了。法国人可以将 70 写成soixante dix，即“六十十”，但尽管他们可以将 65 写成 65，却不能将 610 写成 610。这个符号已被抢先用来表示其他含义。
Ken was a fifth grader who added 35 and 35 and got 610. His bug was showing clearly. Since 32 plus 32 is 64, then 35 plus 35 should be 610. Ken was brought into a better relationship with mathematics when he learned to see his mistake as a trick that mathematical formalisms play on us. The French can say seventy as soixante dix, “sixty-ten,” but although they can write sixty-five as 65, they cannot write sixty-ten as 610. This symbol has been preempted to mean something else.
肯可能表面上看起来对数字的直觉很差。但这种诊断是完全错误的。当被问到“如果你有 35 美元，你又得到了 35 美元，你会有 610 美元吗？”时，他的回答是肯定的“不可能”。当被问到他会有多少钱时，他回到纸上计算，从 610 上划掉零，得出了新的答案 61，直觉上这个答案并不算太离谱。他的问题不是直觉或数字概念不好。从计算机专家的角度来看，可以发现几个困难，每个困难都是可以理解和纠正的。
Ken might superficially appear to have had bad intuitions about numbers. But this is quite wrong as a diagnosis. When asked, “If you had thirty-five dollars and you got thirty-five dollars more, would you have $610.00,” his answer was an emphatic, “No way.” When asked how much he would have, he returned to his paper calculation, crossed off the zero from 610, and came up with the new answer of 61, which intuitively is not so far off. His problem is not bad intuition or notion of number. From a computerist’s point of view one can recognize several difficulties, each of which is understandable and correctable.
首先，他将程序的操作与他的一般知识储备分离开来。更好的程序应该内置“错误检查”。既然他可以在提示时识别错误，那么他当然应该能够设置程序以包括提示自己。其次，当他发现错误时，他没有改变，甚至没有看程序，而只是改变了答案。第三，我对肯的了解告诉我他为什么不试图改变程序。在发生这件事时，他没有将程序视为实体，视为可以命名、操纵或更改的事物。因此，修复他的程序确实远远超出了他的意识。程序是可以调试的东西，这一想法对许多孩子来说是一个强大而困难的概念，直到他们积累了使用它们的经验。
First, he dissociates the operation of the procedure from his general store of knowledge. A better procedure would have an “error check” built into it. Since he could recognize the error when prompted, he certainly should have been capable of setting up the procedure to include prompting himself. Second, when he found the error he did not change, or even look at, the procedure, but merely changed the answer. Third, my knowledge of Ken tells me why he did not try to change the procedure. At the time of this incident he did not recognize procedures as entities, as things one could name, manipulate, or change. Thus, fixing his procedures is very far indeed from his awareness. The idea of procedures as things that can be debugged is a powerful, difficult concept for many children, until they have accumulated experience in working with them.
我见过像肯这样的孩子在LOGO环境中编写程序后克服了这种困难。但是为什么孩子们不从日常生活中学习程序方法呢？每个人在日常生活中都会使用程序。玩游戏或为迷路的驾驶员指路都是程序思维的练习。但在日常生活中，程序是生活和使用的，它们不一定会被反思。在LOGO环境中，随着孩子们逐渐掌握程序的概念，程序变成了一个被命名、操纵和识别的东西。对于像肯这样的人来说，这种影响是，日常生活中的程序和编程经验现在成为学校进行正式算术的资源。当我们使用计算隐喻将牛顿运动定律与更个人化和概念上更强大的事物联系起来时，牛顿运动定律就变得生动起来。当我们将几何学与最基本的人类经验中的前身联系起来时，几何学就变得生动起来：一个人在空间中的体验。同样，当我们能够为个体学习者建立与程序前身的联系时，正式算术也会变得生动起来。这些前体确实存在。孩子确实拥有程序性知识，而且确实在生活的很多方面运用这些知识，无论是制定井字游戏的策略，还是为迷路的司机指路。但很多时候，同一个孩子在学校的算术课上却不会运用这些知识。
I have seen children like Ken get over this kind of difficulty after some experience writing programs in a LOGO environment. But why don’t children learn a procedural approach from daily life? Everyone works with procedures in everyday life. Playing a game or giving directions to a lost motorist are exercises in procedural thinking. But in everyday life procedures are lived and used, they are not necessarily reflected on. In the LOGO environment, a procedure becomes a thing that is named, manipulated, and recognized as the children come to acquire the idea of procedure. The effect of this for someone like Ken is that everyday-life experience of procedures and programming now becomes a resource for doing formal arithmetic in school. Newton’s laws of motion came alive when we used computational metaphor to tie them to more personal and conceptually powerful things. Geometry came alive when we connected it to its precursors in the most fundamental human experience: the experience of one’s body in space. Similarly, formal arithmetic will come alive when we can develop links for the individual learner with its procedural precursors. And these precursors do exist. The child does have procedural knowledge and he does use it in many aspects of his life, whether in planning strategies for a game of tic-tac-toe or in giving directions to a motorist who has lost his way. But all too often the same child does not use it in school arithmetic.
这种情况与我们在ARI和GAL的对话中遇到的情况完全一样，也与使用海龟圈模型改变对绳子和硬币问题产生的循环直觉的情况完全一样。在所有这些情况下，我们感兴趣的是如何将一个强大的想法变成直觉思维的一部分。我不知道如何培养孩子关于何时以及如何使用程序性想法的直觉，但我认为我们能做的最好的事情就是了解一个新人的比喻所暗示的。作为教育工作者，我们可以通过为儿童创造条件来有效和愉快地使用程序性思维来提供帮助。我们可以通过让他们接触许多与程序性有关的概念来提供帮助。这是通过LOGO环境的概念内容实现的。
The situation is exactly like the one we met in the dialog between ARI and GAL and in the use of the Turtle circle model to change the intuition of circularity brought to bear on the string and coin problems. In all these cases, we are interested in how a powerful idea is made part of intuitive thinking. I do not know a recipe for developing a child’s intuition about when and how to use procedural ideas, but I think that the best we can do is what is suggested by the metaphor of getting to know a new person. As educators we can help by creating the conditions for children to use procedural thinking effectively and joyfully. And we can help by giving them access to many concepts related to procedurality. This is achieved through the conceptual content of LOGO environments.
在本书中，我明确指出程序性思维是一种强大的智力工具，甚至建议将自己比作计算机作为实现这一目标的策略。人们常常担心使用计算机模型来模拟人类会导致机械或线性思维：他们担心人们会失去对直觉、价值观和判断力的尊重。他们担心工具理性会成为良好思维的典范。我认真对待这些担忧，但并不认为它们是对计算机本身的担忧，而是对文化如何吸收计算机存在的担忧。“像计算机一样思考”这一建议可以理解为总是像计算机一样思考一切。这将是限制性的和狭隘的。但这个建议可以从一个完全不同的意义上来理解，它不是排除任何东西，而是为一个人的思维工具库增添强大的补充。作为回报，什么都没有放弃。认为一个人必须放弃旧方法才能采用新方法，这意味着一种在我看来幼稚且没有根据的人类心理学理论。在我看来，人类智能的一个显著特征是能够运用多种认知方式，通常是并行的，这样就可以在多个层面上理解某件事。根据我的经验，我要求自己“像计算机一样思考”这一事实并不会封闭其他认识论。它只是为思考开辟了新的方式。文化对计算机的同化将产生计算机素养。这句话通常被理解为知道如何编程，或者了解计算机的各种用途。但真正的计算机素养不仅仅是知道如何使用计算机和计算思想。它还知道什么时候这样做是合适的。
In this book I have clearly been arguing that procedural thinking is a powerful intellectual tool and even suggested analogizing oneself to a computer as a strategy for doing it. People often fear that using computer models for people will lead to mechanical or linear thinking: They worry about people losing respect for their intuitions, sense of values, powers of judgment. They worry about instrumental reason becoming a model for good thinking. I take these fears seriously but do not see them as fears about computers themselves but rather as fears about how culture will assimilate the computer presence. The advice “think like a computer” could be taken to mean always think about everything like a computer. This would be restrictive and narrowing. But the advice could be taken in a much different sense, not precluding anything, but making a powerful addition to a person’s stock of mental tools. Nothing is given up in return. To suggest that one must give up an old method in order to adopt a new one implies a theory of human psychology that strikes me as naive and unsupported. In my view a salient feature of human intelligence is the ability to operate with many ways of knowing, often in parallel, so that something can be understood on many levels. In my experience, the fact that I ask myself to “think like a computer” does not close off other epistemologies. It simply opens new ways for approaching thinking. The cultural assimilation of the computer presence will give rise to a computer literacy. This phrase is often taken as meaning knowing how to program, or knowing about the varied uses made of computers. But true computer literacy is not just knowing how to make use of computers and computational ideas. It is knowing when it is appropriate to do so.
第七章
CHAPTER 7
LOGO 的根源
LOGO’s Roots
皮亚杰与人工智能
Piaget and AI
读者已经遇到了各种各样的学习情况，这些情况是由一套关于如何实现有效学习的共同思想所引导的。在本章中，我们将直接讨论这些思想以及它们所依据的理论来源。其中我们重点关注两个：第一，皮亚杰的影响；第二，计算理论和人工智能的影响。
THE READER HAS ALREADY MET A VARIETY OF LEARNING SITUATIONS drawn together by a common set of ideas about what makes for effective learning. In this chapter we turn directly to these ideas and to the theoretical sources by which they are informed. Of these we focus on two: first, the Piagetian influence, and second, the influence of computational theory and artificial intelligence.
我以前曾谈到过“皮亚杰学习”，即人们在与环境互动中自然、自发的学习，并将其与传统学校的课程驱动学习特征进行了对比。但皮亚杰对我的工作的贡献要深刻得多，理论性和哲学性更强。在本章中，我将介绍一个与大多数人所期待的皮亚杰截然不同的皮亚杰。我不会谈论阶段，也不会强调某个年龄段的孩子能学什么或不能学什么。相反，我将关注作为认识论者的皮亚杰，因为他的思想为我一直在描述的知识型学习理论做出了贡献，这种理论并没有将数学学习方式的研究与数学本身的研究分开。
I have previously spoken of “Piagetian learning,” the natural, spontaneous learning of people in interaction with their environment, and contrasted it with the curriculum-driven learning characteristic of traditional schools. But Piaget’s contribution to my work has been much deeper, more theoretical and philosophical. In this chapter I will present a Piaget very different from the one most people have come to expect. There will be no talk of stages, no emphasis on what children at certain ages can or cannot learn to do. Rather I shall be concerned with Piaget the epistemologist, as his ideas have contributed toward the knowledge-based theory of learning that I have been describing, a theory that does not divorce the study of how mathematics is learned from the study of mathematics itself.
我认为皮亚杰思想的这些认识论方面一直被低估，因为到目前为止，它们还没有为传统教育领域提供任何行动的可能性。但在计算机丰富的教育环境中，即未来十年的教育环境中，情况将不会如此。在第 5 章和海龟思想本身的发展中，我们看到了对数学领域（微分系统数学）的基本原理进行认识论探究的例子，这些探究已经在具体有效的教育设计中取得了成果。阶段理论中的皮亚杰本质上是保守的，几乎是反动的，他强调孩子们不能做什么。我努力发现一个更具革命性的皮亚杰，他的认识论思想可能会扩展人类思维的已知界限。这么多年来，他们一直无法做到这一点，因为缺乏实现手段，而数学计算机现在开始提供这种技术。
I think these epistemological aspects of Piaget’s thought have been underplayed because up until now they offered no possibilities for action in the world of traditional education. But in a computer-rich educational environment, the educational environment of the next decade, this will not be the case. In chapter 5 and in the development of the Turtle idea itself, we saw examples of how an epistemological inquiry into what is fundamental in a sector of mathematics, the mathematics of differential systems, has already paid off in concrete, effective educational designs. The Piaget of the stage theory is essentially conservative, almost reactionary, in emphasizing what children cannot do. I strive to uncover a more revolutionary Piaget, one whose epistemological ideas might expand known bounds of the human mind. For all these years they could not do so for lack of a means of implementation, a technology which the mathetic computer now begins to make available.
本章所介绍的皮亚杰在另一种意义上也是新的。他被置于一个理论框架中，这个框架来自计算机世界的一个方面，我们虽然没有直接谈到这个方面，但它的观点一直隐含在本书的始终。人工智能的定义可狭义可广义。从狭义上讲，人工智能涉及扩展机器执行如果由人来执行则会被视为智能的功能的能力。它的目标是构造机器，在这样做的过程中，它可以被视为高级工程的一个分支。但为了构造这样的机器，通常不仅需要反思机器的性质，还需要反思要执行的智能功能的性质。
The Piaget as presented in this chapter is new in another sense as well. He is placed in a theoretical framework drawn from a side of the computer world of which we have not spoken directly, but whose perspectives have been implicit throughout this book, that of artificial intelligence, or AI. The definition of artificial intelligence can be narrow or broad. In the narrow sense, AI is concerned with extending the capacity of machines to perform functions that would be considered intelligent if performed by people. Its goal is to construct machines, and, in doing so, it can be thought of as a branch of advanced engineering. But in order to construct such machines, it is usually necessary to reflect not only on the nature of machines but on the nature of the intelligent functions to be performed.
例如，要制造一台可以用自然语言进行指导的机器，就必须深入探究语言的本质。为了制造一台能够学习的机器，我们必须深入探究学习的本质。从这种研究中可以得出人工智能的更广泛定义：认知科学。从这个意义上讲，人工智能与语言学和心理学等较古老的学科有相同的领域。但人工智能的独特之处在于，它的方法论和理论风格在很大程度上借鉴了计算理论。在本章中，我们将以多种方式使用这种理论风格：首先，重新诠释皮亚杰；其次，发展学习和理解理论，为我们的教育情境设计提供指导；第三，以一种更不寻常的方式。人工智能的目的是将以前可能看似抽象甚至形而上学的思维思想具体化。正是这种具体化的特性使人工智能的思想对许多当代心理学家如此有吸引力。我们建议向儿童教授人工智能，以便他们也能更具体地思考心理过程。心理学家利用人工智能的理念构建关于心理过程的正式科学理论，而儿童则以更非正式和个人化的方式使用相同的理念来思考自己。显然，我认为这是一件好事，因为表达思维过程的能力使我们能够改进思维过程。
For example, to make a machine that can be instructed in natural language, it is necessary to probe deeply into the nature of language. In order to make a machine capable of learning, we have to probe deeply into the nature of learning. And from this kind of research comes the broader definition of artificial intelligence: that of a cognitive science. In this sense, AI shares its domain with the older disciplines such as linguistics and psychology. But what is distinctive in AI is that its methodology and style of theorizing draw heavily on theories of computation. In this chapter we shall use this style of theorizing in several ways: first, to reinterpret Piaget; second, to develop the theories of learning and understanding that inform our design of educational situations; and third, in a somewhat more unusual way. The aim of AI is to give concrete form to ideas about thinking that previously might have seemed abstract, even metaphysical. It is this concretizing quality that has made ideas from AI so attractive to many contemporary psychologists. We propose to teach AI to children so that they, too, can think more concretely about mental processes. While psychologists use ideas from AI to build formal, scientific theories about mental processes, children use the same ideas in a more informal and personal way to think about themselves. And obviously I believe this to be a good thing in that the ability to articulate the processes of thinking enables us to improve them.
皮亚杰曾自称是一位认识论者。他这样说是什么意思呢？当他谈到儿童的发展时，他实际上谈论的是知识的发展。这句话使我们对认识论和心理学对学习的理解方式进行了对比。从心理学的角度来看，重点是学习者所遵循的规律，而不是学习的内容。行为主义者研究强化计划，动机理论家研究驱动力，格式塔理论家研究良好形式。对皮亚杰来说，将学习过程与学习内容分开是一个错误。要了解儿童如何学习数字，我们必须研究数字。我们必须以一种特殊的方式研究数字：我们必须研究数字的结构，这是一项严肃的数学工作。这就是为什么皮亚杰在同一段话中提到幼儿行为和理论数学家的关注点并不罕见的原因。为了使通过关注学习内容的结构来研究学习的想法更加具体，我们来看一下日常生活中非常具体的学习内容，看看它从心理学和认识论的角度有多么不同。
Piaget has described himself as an epistemologist. What does he mean by that? When he talks about the developing child, he is really talking as much about the development of knowledge. This statement leads us to a contrast between epistemological and psychological ways of understanding learning. In the psychological perspective, the focus is on the laws that govern the learner rather than on what is being learned. Behaviorists study reinforcement schedules, motivation theorists study drive, gestalt theorists study good form. For Piaget, the separation between the learning process and what is being learned is a mistake. To understand how a child learns number, we have to study number. And we have to study number in a particular way: We have to study the structure of number, a mathematically serious undertaking. This is why it is not at all unusual to find Piaget referring in one and the same paragraph to the behavior of small children and to the concerns of theoretical mathematicians. To make more concrete the idea of studying learning by focusing on the structure of what is learned, we look at a very concrete piece of learning from everyday life and see how different it appears from a psychological and from an epistemological perspective.
我们将考虑学习骑自行车。如果我们不了解更多，骑自行车似乎是一件非常了不起的事情。是什么使它成为可能？人们可以通过研究骑车人来探究这个问题，找出哪些特殊属性（反应速度、大脑功能的复杂性、动机的强度）有助于他的表现。这个问题虽然很有趣，但与问题的真正解决方案无关。人们可以骑自行车，因为自行车一旦运动起来，就具有内在稳定性。没有骑手的自行车在陡峭的下坡上推下不会翻倒；它会无限地滑下山坡。前叉的几何结构确保如果自行车向左倾斜，车轮将向左旋转，从而导致自行车转弯并产生离心力，将自行车向右抛出，抵消跌倒的趋势。没有骑手的自行车保持着完美的平衡。对于新手来说，它会摔倒。这是因为新手对平衡的直觉是错误的，会僵化自行车的位置，导致自行车自身的矫正机制无法自由运作。因此，学习骑车并不意味着学习平衡，而是学习不失衡，学习不干扰。
We will consider learning to ride a bicycle. If we did not know better, riding a bicycle would seem to be a really remarkable thing. What makes it possible? One could pursue this question by studying the rider to find out what special attributes (speed of reaction, complexity of brain functioning, intensity of motivation) contribute to his performance. This inquiry, interesting though it might be, is irrelevant to the real solution to the problem. People can ride bicycles because the bicycle, once in motion, is inherently stable. A bicycle without a rider pushed off on a steep downgrade will not fall over; it will run indefinitely down the hill. The geometrical construction of the front fork ensures that if the bicycle leans to the left the wheel will rotate to the left, thus causing that bicycle to turn and produce a centrifugal force that throws the bicycle to the right, counteracting the tendency to fall. The bicycle without a rider balances perfectly well. With a novice rider it will fall. This is because the novice has the wrong intuitions about balancing and freezes the position of the bicycle so that its own corrective mechanism cannot work freely. Thus learning to ride does not mean learning to balance, it means learning not to unbalance, learning not to interfere.
我们在这里所做的是通过深入了解所学内容来理解学习过程。心理学原理与此无关。就像我们通过研究自行车来了解人们如何骑自行车一样，皮亚杰教导我们，我们应该通过更深入地了解数字是什么来了解孩子如何学习数字。
What we have done here is understand a process of learning by acquiring deeper insight into what was being learned. Psychological principles had nothing to do with it. And just as we have understood how people ride bicycles by studying bicycles, Piaget has taught us that we should understand how children learn number through a deeper understanding of what number is.
对数字的本质感兴趣的数学家从不同的角度看待这个问题。一种与形式主义者有关的方法，试图通过建立公理来理解数字。第二种与伯特兰·罗素有关的方法，试图通过将数字简化为更基本的东西来定义数字，例如逻辑和集合论。虽然这两种方法都是数学史上有效且重要的篇章，但它们都没有阐明数字为什么是可以学习的。但有一个数学学派确实这样做了，尽管这不是它的本意。这就是布尔巴基学派的结构主义。1布尔巴基是一群法国数学家的化名，他们着手阐明数学的统一理论。数学应该是一门学科，而不是一个拥有自己语言和发展路线的分支学科的集合。该学派通过识别一些被称为“母结构”的构建块朝着这个方向发展。这些结构与我们对微观世界的想法有一些共同之处。想象一个微观世界，其中的事物可以有序排列，但没有其他属性。按照布尔巴基学派的说法，如何运作世界的知识是秩序的母结构。第二个微观世界允许邻近关系，这是拓扑的母结构。第三个与组合实体以产生新实体有关；这是代数微观结构。布尔巴基学派统一数学是通过将更复杂的结构（如算术）视为更简单结构的组合来实现的，其中最重要的是三个母结构。这个学派无意提出学习理论。他们希望他们的结构分析成为数学家在日常工作中使用的技术工具。但母结构理论是一种学习理论。它是一种关于数字如何学习的理论。通过展示如何将算术结构分解为更简单但仍然有意义和连贯的结构，数学家们展示了一条通往数值知识的数学途径。毫不奇怪的是，皮亚杰当时明确地寻找一种可以解释儿童数字发展过程的数字理论，并发展了一套类似的、平行的结构，然后在“发现”布尔巴基学派后，能够利用其结构来阐述他自己的结构。
Mathematicians interested in the nature of number have looked at the problem from different standpoints. One approach, associated with the formalists, seeks to understand number by setting up axioms to capture it. A second approach, associated with Bertrand Russell, seeks to define number by reducing it to something more fundamental, for example, logic and set theory. Although both of these approaches are valid, important chapters in the history of mathematics, neither casts light on the question of why number is learnable. But there is a school of mathematics that does do so, although this was not its intention. This is the structuralism of the Bourbaki school.1 Bourbaki is a pseudonym taken by a group of French mathematicians who set out to articulate a uniform theory for mathematics. Mathematics was to be one, not a collection of subdisciplines each with its own language and line of development. The school moved in this direction by recognizing a number of building blocks that it called the “mother structures.” These structures have something in common with our idea of microworlds. Imagine a microworld in which things can be ordered but have no other properties. The knowledge of how to work the world is, in terms of the Bourbaki school, the mother structure of order. A second microworld allows relations of proximity, and this is the mother structure of topology. A third has to do with combining entities to produce new entities; this is the algebraic microstructure. The Bourbaki school’s unification of mathematics is achieved by seeing more complex structures, such as arithmetic, as combinations of simpler structures of which the most important are the three mother structures. This school had no intention of making a theory of learning. They intended their structural analysis to be a technical tool for mathematicians to use in their day-to-day work. But the theory of mother structures is a theory of learning. It is a theory of how number is learnable. By showing how the structure of arithmetic can be decomposed into simpler, but still meaningful and coherent, structures, the mathematicians are showing a mathetic pathway into numerical knowledge. It is not surprising that Piaget, who was explicitly searching for a theory of number that would explain its development in children, developed a similar, parallel set of constructs, and then, upon “discovering” the Bourbaki school was able to use its constructs to elaborate his own.
皮亚杰观察到，儿童发展出连贯的智力结构，这些结构似乎与布尔巴基母结构非常接近。例如，回想一下布尔巴基秩序结构；事实上，从最早的年龄开始，儿童就开始发展对事物进行排序的专业知识。拓扑和代数母结构具有相似的发展前兆。是什么使它们变得可学？首先，每个母结构都代表了儿童生活中的连贯活动，原则上可以独立于其他活动进行学习和理解。
Piaget observed that children develop coherent intellectual structures that seemed to correspond very closely to the Bourbaki mother structures. For example, recall the Bourbaki structure of order; indeed, from the earliest ages, children begin to develop expertise in ordering things. The topological and algebraic mother structures have similar developmental precursors. What makes them learnable? First of all, each represents a coherent activity in the child’s life that could in principle be learned and made sense of independent of the others.
其次，每个知识结构都具有一种内在的简单性，皮亚杰在他的群体理论中对此进行了阐述，后面将以略有不同的术语进行讨论。第三，尽管这些母结构是独立的，但它们是并行学习的，并且它们有共同的形式主义，这些事实表明它们是相互支持的；学习一个知识结构会促进学习其他知识结构。
Second, the knowledge structure of each has a kind of internal simplicity that Piaget has elaborated in his theory of groupements, and which will be discussed in slightly different terms later. Third, although these mother structures are independent, the fact that they are learned in parallel and that they share a common formalism are clues that they are mutually supportive; the learning of each facilitates the learning of the others.
皮亚杰利用这些思想，以儿童头脑中一系列连贯、有规律的结构来解释各种知识领域的发展。他将这些内部结构描述为始终与外部世界相互作用，但他的理论重点一直是内部事件。我的观点更偏向干预主义。我的目标是教育，而不仅仅是理解。因此，在我的思考中，我更加强调皮亚杰作品中隐含但未详述的两个维度：对可能发展的知识结构的兴趣，而不是儿童目前实际发展的知识结构，以及与这些结构产生共鸣的学习环境的设计。乌龟可以用来说明这两种兴趣：首先，识别一组强大的数学思想，我们假设这些思想在儿童身上没有表现出来，至少没有以发展的形式表现出来；其次，创建一个过渡对象，即乌龟，它可以存在于儿童的环境中并与这些思想接触。作为一名数学家，我知道科学史上最强大的思想之一就是微分分析。从牛顿开始，局部与整体之间的关系很好地确定了数学的议程。然而，这个想法在儿童世界中没有立足之地，主要是因为传统上对它的接触依赖于正式的数学训练基础设施。对于大多数人来说，数学中最先进的思想对儿童来说是无法理解的，这是再自然不过的事情了。从我从皮亚杰那里学到的观点来看，我们希望找到联系。所以我们开始寻找一些联系。但找到联系并不意味着发明一种新的聪明的“激励”教学法。它意味着一个研究议程，包括将微分思想中最强大的部分与无法理解的形式主义的偶然性区分开来。然后目标是将这些科学上的基本结构与心理上强大的结构联系起来。当然，这些是海龟圈、物理微观世界和触摸传感器海龟的基础思想。
Piaget has used these ideas to give an account of the development of a variety of domains of knowledge in terms of a coherent, lawful set of structures as processes within the child’s mind. He describes these internal structures as always in interaction with the external world, but his theoretical emphasis has been the internal events. My perspective is more interventionist. My goals are education, not just understanding. So, in my own thinking I have placed a greater emphasis on two dimensions implicit but not elaborated in Piaget’s own work: an interest in intellectual structures that could develop as opposed to those that actually at present do develop in the child, and the design of learning environments that are resonant with them. The Turtle can be used to illustrate both of these interests: first, the identification of a powerful set of mathematical ideas that we do not presume to be represented, at least not in a developed form, in children; second, the creation of a transitional object, the Turtle, that can exist in the child’s environment and make contact with the ideas. As a mathematician I know that one of the most powerful ideas in the history of science was that of differential analysis. From Newton onward, the relationship between the local and the global pretty well set the agenda for mathematics. Yet this idea has had no place in the world of children, largely because traditional access to it depends on an infrastructure of formal, mathematical training. For most people, nothing is more natural than that the most advanced ideas in mathematics should be inaccessible to children. From the perspective I took from Piaget, we would expect to find connections. So we set out to find some. But finding the connections did not simply mean inventing a new kind of clever, “motivating” pedagogy. It meant a research agenda that included separating what was most powerful in the idea of differential from the accidents of inaccessible formalisms. The goal was then to connect these scientifically fundamental structures with psychologically powerful ones. And of course these were the ideas that underlay the Turtle circle, the physics microworlds, and the touch-sensor Turtle.
从什么意义上说，自然环境是微观世界的来源，实际上是微观世界网络的来源？让我们将整个自然环境缩小到其中可能充当一个特定微观世界的来源的事物，即配对的、一一对应的微观世界。孩子们看到的很多东西都是成对的：母亲和父亲、刀和叉、鸡蛋和蛋杯。他们也被要求成为成对的积极构建者。他们被要求分类袜子、为每个人摆放一个餐具的桌子，并分发糖果。当孩子们将注意力集中在成对的东西上时，他们就处于一个自我构建的微观世界，一个成对的微观世界，就像我们将学生置于几何和物理海龟的微观世界中一样。在这两种情况下，相关的微观世界都摆脱了复杂性，变得简单、易于理解。在这两种情况下，孩子都可以自由地玩弄其中的元素。虽然材料有限制，但组合的探索却没有限制。在这两种情况下，环境的力量都在于它“富有发现”。
In what sense is the natural environment a source of microworlds, indeed a source for a network of microworlds? Let’s narrow the whole natural environment to those things in it that may serve as a source for one specific microworld, a microworld of pairing, of one-to-one correspondence. Much of what children see comes in pairs: mothers and fathers, knives and forks, eggs and egg cups. And they, too, are asked to be active constructors of pairs. They are asked to sort socks, lay the table with one place setting for each person, and distribute candies. When children focus attention on pairs they are in a self-constructed microworld, a microworld of pairs, in the same sense as we placed our students in the microworlds of geometry and physics Turtles. In both cases the relevant microworld is stripped of complexity, is simple, graspable. In both cases the child is allowed to play freely with its elements. Although there are constraints on the materials, there are no constraints on the exploration of combinations. And in both cases the power of the environment is that it is “discovery rich.”
使用计算机可以更清楚地看到孩子们构建自己的个人微观世界。第 4 章末尾的黛博拉的故事就是一个很好的例子。LOGO让她有机会构建一个特别整洁的微观世界，即她的“ RIGHT 30 世界”。但她可能在没有计算机的情况下在脑海中完成了类似的事情。例如，她可能决定以一组简单的操作来理解现实世界中的方向。这种智力事件通常对观察者来说是看不见的，就像我的代数老师不知道我用齿轮来思考方程式一样。但如果仔细观察，就可以看到它们。麻省理工学院LOGO小组成员罗伯特·劳勒 (Robert Lawler) 在他的博士研究中最清楚地证明了这一点。劳勒开始观察一个六岁孩子，他的女儿米里亚姆在六个月内所做的一切。他获得的大量信息使他能够拼凑出米里亚姆成长能力的微观结构的画面。例如，在这段时间里，米丽亚姆学会了加法，而劳勒能够证明，这并不包括习得一个逻辑上统一的程序。一个更好的模型是，她将许多独特的微观世界带入了工作关系中，每个微观世界都可以追溯到可识别的、以前的经历。
Working with computers can make it more apparent that children construct their own personal microworlds. The story of Deborah at the end of chapter 4 is a good example. LOGO gave her the opportunity to construct a particularly tidy microworld, her “RIGHT 30 world.” But she might have done something like this in her head without a computer. For example, she might have decided to understand directions in the real world in terms of a simple set of operations. Such intellectual events are not usually visible to observers, any more than my algebra teachers knew that I used gears to think about equations. But they can be seen if one looks closely enough. Robert Lawler, a member of the Massachusetts Institute of Technology LOGO group, demonstrated this most clearly in his doctoral research. Lawler set out to observe everything a six-year-old child, his daughter Miriam, did during a six-month period. The wealth of information he obtained allowed him to piece together a picture of the microstructure of Miriam’s growing abilities. For example, during this period Miriam learned to add, and Lawler was able to show that this did not consist of acquiring one logically uniform procedure. A better model of her learning to add is that she brought into a working relationship a number of idiosyncratic microworlds, each of which could be traced to identifiable, previous experiences.
我说过皮亚杰是一位认识论者，但没有详细阐述他是一位怎样的认识论者。认识论是知识的理论。根据其词源，认识论一词可用于涵盖有关知识的所有知识，但传统上它被用在一种相当特殊的方式中：即描述对知识有效性条件的研究。皮亚杰的认识论并不关心知识的有效性，而是关心知识的起源和发展。他关心知识的起源和演化，并将他的研究领域描述为“发生认识论”，以此来表明这一事实。传统的认识论经常被视为哲学的一个分支。发生认识论致力于将自己定位为一门科学。它的研究者收集数据并发展关于知识如何发展的理论，有时关注历史中知识的演化，有时关注个人知识的演化。但它并不认为这两个领域是截然不同的：它试图理解它们之间的关系。这些关系可以采取不同的形式。
I have said that Piaget is an epistemologist, but have not elaborated on what kind. Epistemology is the theory of knowledge. The term epistemology could, according to its etymology, be used to cover all knowledge about knowledge, but traditionally it has been used in a rather special way: that is, to describe the study of the conditions of validity of knowledge. Piaget’s epistemology is concerned not with the validity of knowledge but with its origin and growth. He is concerned with the genesis and evolution of knowledge, and marks this fact by describing his field of study as “genetic epistemology.” Traditional epistemology has often been taken as a branch of philosophy. Genetic epistemology works to assert itself as a science. Its students gather data and develop theories about how knowledge developed, sometimes focusing on the evolution of knowledge in history, sometimes on the evolution of knowledge in the individual. But it does not see the two realms as distinct: It seeks to understand relations between them. These relations can take different forms.
在最简单的情况下，个体发展与历史发展是平行的，回想一下生物学家的名言，个体发生重演了系统发生。例如，儿童一律以亚里士多德的方式代表物理世界，例如，他们认为力作用于位置而不是速度。在其他情况下，这种关系更为复杂，甚至到了逆转的地步。儿童发展中首先出现的智力结构有时不是早期科学的特征，而是最现代科学的特征。因此，例如，母结构拓扑学在儿童发展的早期就出现了，但拓扑学本身只是在现代才作为数学分支学科出现。只有当数学变得足够先进时，它才能发现自己的起源。
In the simplest case the individual development is parallel to the historical development, recalling the biologists’ dictum, ontogeny recapitulates phylogeny. For example, children uniformly represent the physical world in an Aristotelian manner, thinking, for example, that forces act on position rather than on velocities. In other cases, the relation is more complex, indeed to the point of reversal. Intellectual structures that appear first in a child’s development are sometimes characteristic not of early science but of the most modern science. So, for example, the mother structure topology appears very early in the child’s development, but topology itself appeared as a mathematical subdiscipline only in modern times. Only when mathematics becomes sufficiently advanced is it able to discover its own origins.
在二十世纪早期，形式逻辑被视为数学基础的同义词。直到布尔巴基的结构主义理论出现，我们才看到数学内部的发展，从而让该领域能够“记住”其遗传根源。通过发生认识论的工作，这种“记忆”使数学与关于儿童如何构建现实的研究发展建立了最密切的关系。
In the early part of the twentieth century, formal logic was seen as synonymous with the foundation of mathematics. Not until Bourbaki’s structuralist theory appeared do we see an internal development in mathematics that opened the field up to “remembering” its genetic roots. And through the work of genetic epistemology, this “remembering” puts mathematics in the closest possible relationship to the development of research about how children construct their reality.
发生认识论已开始主张知识结构与为掌握这些知识而产生的心智结构之间存在一套同源性。布尔巴基的母结构不仅仅是构成数字概念的元素；相反，在心智为自己构建数字时，也存在同源性。因此，研究知识结构的重要性不仅在于更好地理解知识本身，还在于了解人。
Genetic epistemology has come to assert a set of homologies between the structures of knowledge and the structures of the mind that come into being to grasp this knowledge. Bourbaki’s mother structures are not simply the elements that underly the concept of number; rather, homologies are found in the mind as it constructs number for itself. Thus, the importance of studying the structure of knowledge is not just to better understand the knowledge itself, but to understand the person.
对这一辩证过程结构的研究转化为这样一种信念：无论是人还是知识（包括数学），都无法脱离彼此而完全掌握，沃伦·麦卡洛克雄辩地表达了这一信念，他与诺伯特·维纳一起创立了控制论。当麦卡洛克年轻时被问及什么问题将指导他的科学生涯时，他回答道：“是什么造就了人类能够理解数字，又是什么造就了数字能够让人理解它？”
Research on the structure of this dialectical process translates into the belief that neither people nor knowledge—including mathematics—can be fully grasped separately from the other, a belief that was eloquently expressed by Warren McCulloch, who, together with Norbert Wiener, should have credit for founding cybernetics. When asked, as a youth, what question would guide his scientific life, McCulloch answered: “What is a man so made that he can understand number and what is number so made that a man can understand it?”
对于麦卡洛克和皮亚杰来说，对人的研究和对他们所学和所想的研究是密不可分的。对某些人来说，也许有些自相矛盾的是，对这种密不可分的关系的本质的研究已经通过对机器及其所体现的知识的研究而得到推进。我们现在要讨论的就是这种研究方法，即人工智能的研究方法。
For McCulloch as for Piaget, the study of people and the study of what they learn and think are inseparable. Perhaps paradoxically for some, research on the nature of that inseparable relationship has been advanced by the study of machines and the knowledge they can embody. And it is to this research methodology, that of artificial intelligence, that we now turn.
在人工智能领域，研究人员使用计算模型来深入了解人类心理学，并反思人类心理学，以此作为如何使机器模拟人类智能的思想来源。许多人认为这种做法不合逻辑：即使性能看起来相同，有什么理由认为底层过程是相同的？其他人认为这是不合法的：神学和神话都认为人与机器之间的界限是不可改变的。人们担心，如果我们将我们的“判断”与计算机的“计算”进行不恰当的类比，我们会将本质上是人类的东西非人化。我非常认真地对待这些反对意见，但我觉得它们是基于一种比我自己感兴趣的任何东西都更还原论的人工智能观点。一个简短的寓言和一些半开玩笑的类比推理表达了我对这个问题的看法。
In artificial intelligence, researchers use computational models to gain insight into human psychology as well as reflect on human psychology as a source of ideas about how to make mechanisms emulate human intelligence. This enterprise strikes many as illogical: Even when the performance looks identical, is there any reason to think that underlying processes are the same? Others find it illicit: The line between man and machine is seen as immutable by both theology and mythology. There is a fear that we will dehumanize what is essentially human by inappropriate analogies between our “judgments” and those computer “calculations.” I take these objections very seriously, but feel that they are based on a view of artificial intelligence that is more reductionist that anything I myself am interested in. A brief parable and some only half-humored reasoning by analogy express my own views on the matter.
人类一直对飞行很感兴趣。从前，科学家决心了解鸟类如何飞行。他们首先观察鸟类，希望将鸟类翅膀的运动与向上运动联系起来。然后他们进行实验，发现当羽毛被拔掉时，鸟就无法再飞翔。在确定羽毛是飞行器官后，科学家们将精力集中在对羽毛进行微观和超微观研究上，以发现羽毛赋予飞行能力的本质。
Men have always been interested in flying. Once upon a time, scientists determined to understand how birds fly. First they watched them, hoping to correlate the motion of a bird’s wings with its upward movement. Then they proceeded to experiment and found that when its feathers were plucked, a bird could no longer fly. Having thus determined that feathers were the organ of flight, the scientists then focused their efforts on microscopic and ultramicroscopic investigation of feathers in order to discover the nature of their flight-giving power.
事实上，我们目前对鸟类飞行方式的理解并非来自一项只关注鸟类的研究，而且对羽毛的研究毫无收获。相反，这种理解来自对不同种类的现象的研究，需要不同的方法。一些研究涉及理想流体运动定律的高度数学研究。其他研究，最接近我们这里的中心点，包括建造“人工飞行”机器。当然，我们必须将对鸟类飞行的实际观察添加到列表中。所有这些研究活动通过我们对鸟类“自然飞行”和飞机“人工飞行”的理解，协同催生了航空科学。正是本着同样的精神，我认为数学和机器智能方面的各种研究将与心理学协同作用，催生出一门认知科学学科，其原理将适用于自然科学和人工智能。
In reality our current understanding of how birds fly did not come through a study narrowly focused on birds and gained nothing at all from the study of feathers. Rather, it came from studying phenomena of different kinds and requiring different methodologies. Some research involved highly mathematical studies in the laws of motion of idealized fluids. Other research, closest to our central point here, consisted of building machines for “artificial flight.” And, of course, we must add to the list the actual observation of bird flight. All these research activities synergistically gave rise to aeronautical science through what we understand of the “natural flight” of birds and the “artificial flight” of airplanes. And it is in much the same spirit that I imagine diverse investigations in mathematics and in machine intelligence to act synergistically with psychology in giving rise to a discipline of cognitive science whose principles would apply to natural and to artificial intelligence.
将对人工智能的常见反对意见转移到飞行的背景下是有益的。这让我们想象怀疑论者会说：“你们数学家处理的是理想化的流体——真实的大气要复杂得多”，或者“你没有理由认为飞机和鸟类的工作方式相同——鸟类没有螺旋桨，飞机没有羽毛。”但这些批评的前提只是在最肤浅的意义上是正确的：同样的原理（例如伯努利定律）适用于真实流体和理想流体，无论流体流过羽毛还是铝制机翼，它们都适用。
It is instructive to transpose to the context of flying the common objections raised against AI. This leads us to imagine skeptics who would say, “You mathematicians deal with idealized fluids—the real atmosphere is vastly more complicated,” or “You have no reason to suppose that airplanes and birds work the same way—birds have no propellors, airplanes have no feathers.” But the premises of these criticisms are true only in the most superficial sense: the same principle (e.g., Bernoulli’s law) applies to real as well as ideal fluids, and they apply whether the fluid flows over a feather or an aluminum wing.
人工智能“认知研究”分支的研究人员并不认同任何一种思考方式，就像传统心理学家一样。对一些人来说，计算机模型用于将所有思考简化为强大演绎系统的形式运算。亚里士多德成功地用这样一些简单的三段论为人类思维的一小部分制定了演绎规则，例如“如果所有人都是凡人，而苏格拉底是人，那么苏格拉底也是凡人。”在 19 世纪，数学家能够将这种推理扩展到更大但仍然受限制的领域。但只有在计算方法的背景下，才有人认真尝试将演绎逻辑扩展到涵盖所有形式的推理，包括常识推理和类比推理。在人工智能早期，使用这种演绎模型非常流行。然而，近年来，该领域的许多工作者采取了几乎截然相反的策略。新方法并不寻求强大的演绎方法，以便从一般原则中得出令人惊讶的结论，而是假设人们能够思考，只是因为他们可以利用大量特定的知识。我们解决问题的方式往往比我们意识到的要多，因为我们“几乎已经知道答案”。一些研究人员试图通过为程序提供大量知识来使程序变得智能，这样解决问题的大部分工作就变成了从内存中某个地方检索知识。
Workers in the “cognitive studies” branch of AI do not share any one way of thinking about thinking, any more than traditional psychologists do. For some, the computer model is used to reduce all thinking to the formal operations of powerful deductive systems. Aristotle succeeded in formulating the deductive rules for a small corner of human thinking in such simple syllogisms as “If all men are mortal and Socrates is a man, then Socrates is mortal.” In the nineteenth century, mathematicians were able to extend this kind of reasoning to a somewhat larger but still restricted area. But only in the context of computational methods has there been a serious attempt to extend deductive logic to cover all forms of reasoning, including commonsense reasoning and reasoning by analogy. Working with this kind of deductive model was very popular in the early days of AI. In recent years, however, many workers in the field have adopted an almost diametrically opposed strategy. Instead of seeking powerful deductive methods that would enable surprising conclusions to be drawn from general principles, the new approach assumes that people are able to think only because they can draw on larger pools of specific, particular knowledge. More often than we realize, we solve problems by “almost knowing the answer” already. Some researchers try to make programs be intelligent by giving them such quantities of knowledge that the greater part of solving a problem becomes its retrieval from somewhere in the memory.
鉴于我作为数学家和皮亚杰心理学家的背景，我自然而然地对那些可能让我更好地思考强大发展过程的计算模型最感兴趣：获得空间思维和处理大小和数量的能力。竞争对手的方法——演绎法和基于知识的方法——倾向于解决给定知识系统的表现，而该系统的结构（如果不是其内容）是静态的。我感兴趣的发展问题需要一个动态模型来说明知识结构本身如何形成和变化。我相信这些是与教育最相关的模型。
Given my background as a mathematician and Piagetian psychologist, I naturally became most interested in the kinds of computational models that might lead me to better thinking about powerful developmental processes: the acquisition of spatial thinking and the ability to deal with size and quantity. The rival approaches—deductive and knowledge based—tended to address performance of a given intellectual system whose structure, if not whose content, remained static. The kind of developmental questions I was interested in needed a dynamic model for how intellectual structures themselves could come into being and change. I believe that these are the kind of models that are most relevant to education.
我知道描述这种方法的最好方法是给出一个深受计算思想影响的理论样本，该理论可以帮助我们理解一种特定的心理现象：皮亚杰守恒定律。我们记得，六七岁的孩子都认为，当液体从一个容器倒入另一个容器时，液体的量会增加或减少。具体来说，当第二个容器比第一个容器更高更窄时，孩子们一致认为液体的量增加了。然后，就像变魔术一样，在大约相同的年龄，所有孩子都改变了想法：他们现在同样毫不含糊地坚持认为液体的量保持不变。
The best way I know to characterize this approach is to give a sample of a theory heavily influenced by ideas from computation that can help us understand a specific psychological phenomenon: Piagetian conservation. We recall that children up until the age of six or seven believe that a quantity of liquid can increase or decrease when it is poured from one container to another. Specifically, when the second container is taller and narrower than the first, the children unanimously assert that the quantity of liquid has increased. And then, as if by magic, at about the same age, all children change their mind: They now just as unequivocally insist that the amount of liquid remains the same.
关于这一现象是如何发生的，人们提出了许多理论。其中一种理论可能听起来最为熟悉，因为它借鉴了传统的心理学范畴，将前保护主义立场归因于儿童受“外表”的支配。儿童的“理性”无法超越事物“看起来”的样子。感知占主导地位。
Many theories have been advanced for how this could come to pass. One of them, which may sound most familiar because it draws on traditional psychological categories, attributes the preconservationist position to the child’s being dominated by “appearances.” The child’s “reason” cannot override how things “seem to be.” Perception rules.
现在让我们转向另一个理论，这次是受计算方法启发的理论。我们再次提出这个问题：为什么狭窄容器中的身高对孩子来说似乎更大，以及这种情况如何改变？
Let us now turn to another theory, this time one inspired by computational methods. Again we ask the question: Why does height in a narrow vessel seem like more to the child, and how does this change?
我们假设在孩子的头脑中存在三个代理，每个代理都以不同的“简单”方式判断数量。I第一个代理，身高，通过垂直长度来判断液体和其他任何东西的数量。身高是孩子生活中的实际代理。它习惯于通过让孩子们背对背站着来比较他们，并平衡孩子们杯子里可口可乐和巧克力牛奶的数量。我们要强调的是，身高不会做任何像“感知”液体数量这样复杂的事情。相反，它狂热地信奉一个抽象的原则：任何东西越高就越多。
Let us posit the existence of three agents in the child’s mind, each of which judges quantities in a different “simple-minded” way.I The first, Aheight judges the quantity of liquids and of anything else by its vertical extent. Aheight is a practical agent in the life of the child. It is accustomed to comparing children by standing them back to back and of equalizing the quantities of Coca-Cola and chocolate milk in children’s glasses. We emphasize that Aheight does not do anything as complicated as “perceive” the quantity of liquid. Rather, it is fanatically dedicated to an abstract principle: Anything that goes higher is more.
还有第二个代理，称为A width，它通过水平范围进行判断。它不像A height那样“熟练” 。它有机会判断海水中有很多水，但在孩子的心目中，这个原则不如A height “有影响力” 。
There is a second agent, called Awidth, that judges by the horizontal extent. It is not as “practiced” as Aheight. It gets its chance to judge that there is a lot of water in the sea, but in the mind of the child this principle is less “influential” than Aheight.
最后，有一个名为A历史的代理，它说数量是相同的，因为它们曾经是相同的。A历史似乎像一个环保主义者的孩子一样说话，但这是一种错觉。A历史没有理解力，即使确实增加了一些数量，它也会说数量是相同的。
Finally, there is an agent called Ahistory that says that the quantities are the same because once they were the same. Ahistory seems to speak like a conservationist child, but this is an illusion. Ahistory has no understanding and would say the quantity is the same even if some had indeed been added.
在对前保护主义儿童进行的实验中，三个主体各自做出自己的“决定”，并要求采纳。众所周知，身高越高，声音越大。但随着孩子进入下一个阶段，这种情况发生了变化。
In the experiment with the preconservationist child, each of the three agents makes its own “decision” and clamors for it to be adopted. As we know, Aheight’S voice speaks the loudest. But this changes as the child moves on to the next stage.
假设存在代理，则有三种方式可以实现这种变化。高度A和宽度A可以变得更加“老练”，例如，除非所有其他条件都相同，否则高度A会失去自己的资格。这意味着高度A只会根据高度来判断那些横截面相等的事物。其次，特权的“资历”可能会发生变化：历史A可能会成为主导声音。这两种变化模式都不是不可能发生的。但还有第三种模式可以以更简单的方式产生相同的效果。其关键思想是高度A和宽度A通过给出相互矛盾的意见来相互抵消。这个想法很有吸引力（并且接近皮亚杰自己的群体式操作组合概念），但也带来了一些问题。为什么这三个代理不相互抵消，这样孩子就没有任何意见了？这个问题由另一个假设来回答（这与皮亚杰将智力操作者组织成群体的想法有很多共同之处）。如果对代理施加了足够的结构，使得A height和A width彼此之间存在特殊关系，但与A history不存在特殊关系，则中和原则可行。我们已经看到，创建新实体的技术在编程系统中非常有效。这就是我们在此假设的过程。一个新的实体，一个新的代理应运而生。这就是A geom ，它充当A height和A width 的监督者。在A height和A width一致的情况下，A geom会非常“权威”地传递它们的信息。但如果它们不一致，A geom就会被削弱，下属的声音就会被中和。必须强调的是，A geom并不意味着“理解” A height和A width做出决策的原因。geom只知道它们是否一致，如果一致，那么在哪个方向上一致。
There are three ways, given our assumption of the presence of agents, for this change to take place. Aheight, and Awidth could become more “sophisticated,” so that, for example, Aheight would disqualify itself except when all other things are equal. This would mean that Aheight would only step forward to judge by height those things that have equal cross sections. Second, there could be a change in “seniority,” in prerogative: Ahistory could become the dominant voice. Neither of these two modes of change is impossible. But there is a third mode that produces the same effect in a simpler way. Its key idea is that Aheight and Awidth neutralize one another by giving contradictory opinions. The idea is attractive (and close to Piaget’s own concept of grouplike compositions of operations) but raises some problems. Why do all three agents not neutralize one another so that the child would have no opinion at all? The question is answered by a further postulate (which has much in common with Piaget’s idea that intellectual operators be organized into groupements). The principle of neutralization becomes workable if enough structure is imposed on the agents for Aheight and Awidth to be in a special relationship with one another but not with Ahistory. We have seen that the technique of creating a new entity works powerfully in programming systems. And this is the process we postulate here. A new entity, a new agent comes into being. This is Ageom, which acts as the supervisor for Aheight and Awidth. In cases where Aheight and Awidth agree, Ageom passes on their message with great “authority.” But if they disagree, Ageom is undermined and the voices of the underlings are neutralized. It must be emphasized that Ageom is not meant to “understand” the reasons for decision making by Aheight and Awidth. Ageom knows nothing except whether they agree and, if so, in which direction.
这个模型过于简单，它认为即使是儿童思维的简单部分（比如这个守恒）也可以通过四个主体的相互作用来理解。要解释真实过程的复杂性，需要几十个或几百个主体。但是，尽管它很简单，但这个模型准确地传达了该理论的一些原则：特别是，系统的组成部分更像人而不是命题，它们的相互作用更像社会互动而不是数学逻辑的运算。这种观点的转变使我们能够解决发展心理学中的许多技术问题。特别是，我们可以将逻辑学习理解为与社会和身体学习连续进行的。
This model is absurdly oversimplified in suggesting that even so simple a piece of a child’s thinking (such as this conservation) can be understood in terms of interactions of four agents. Dozens or hundreds are needed to account for the complexity of the real process. But, despite its simplicity, the model accurately conveys some of the principles of the theory: in particular, that the components of the system are more like people than they are like propositions and their interactions are more like social interactions than like the operations of mathematical logic. This shift in perspective allows us to solve many technical problems in developmental psychology. In particular, we can understand logical learning as continuous with social and bodily learning.
我说过，这一理论受到计算隐喻的启发。有人可能会问，究竟是如何做到的。这种“理论”似乎不过是拟人化的言论。但我们已经看到，拟人化描述往往是迈向计算理论的一步。心智社会理论的主旨是，代理可以转化为精确的计算模型。只要我们只将这些代理视为“人”，该理论就是循环的。它用人的行为来解释人的行为。但是，如果我们可以将代理视为类似于过程MAN 中的子过程VEE、LINE和HEAD的定义明确的计算实体，那么一切都会变得更加清晰。我们看到，即使在小程序中，非常简单的模块也可以组合在一起产生复杂的结果。
I have said that this theory is inspired by a computational metaphor. One might ask how. The “theory” might appear to be nothing but anthropomorphic talk. But we have already seen that anthropomorphic descriptions are often a step toward computational theories. And the thrust of the society-of-mind theory is that agents can be translated into precise computational models. As long as we only think about these agents as “people,” the theory is circular. It explains the behavior of people in terms of the behavior of people. But, if we can think of the agents as well-defined computational entities similar to the subprocedures VEE, LINE, and HEAD in the procedure MAN, everything becomes clearer. We saw even in small programs how very simple modules can be put together to produce complex results.
这种计算论证使心智社会理论免于依赖恶性循环的指控。但它并不能使它免于循环：相反，就像第 3 章中SPI程序风格的递归程序一样，该理论的大部分力量都来自对“循环逻辑”的建设性使用。传统逻辑学家在研究SPI是如何通过参考SPI来定义的时可能会反对，但计算机程序员和遗传认识论者都认为这种自我参考不仅合法而且必要。他们都认为它具有悖论元素，而这种悖论只能通过注意儿童如何使用他们的“劣等”逻辑来构建他们下一发展阶段的“优越”逻辑来部分捕捉。在皮亚杰漫长的职业生涯中，他越来越强调儿童反思自己思维的能力对智力成长的重要性。“数学悖论”在于这种反思必须来自儿童当前的智力系统。
This computational argument saves the society-of-mind theory from the charge of relying on a vicious circle. But it does not save it from being circular: On the contrary, like recursive programs in the style of the procedure SPI of chapter 3, the theory derives much of its power from a constructive use of “circular logic.” A traditional logician looking at how SPI was defined by reference to SPI might have objected, but the computer programmers and genetic epistemologists share a vision in which this kind of self-reference is not only legitimate but necessary. And both see it as having an element of paradox that is only very partially captured by noting how children use their “inferior” logic to construct the “superior” logic of their next phase of development. To an increasing extent throughout his long career, Piaget has emphasized the importance for intellectual growth of children’s ability to reflect on their own thinking. The “mathetic paradox” lies in the fact that this reflection must be from within the child’s current intellectual system.
尽管四主体保护理论过于简单，几乎是隐喻性的，但它抓住了悖论的一个元素。数理逻辑学家可能想在A 的高度和A 的宽度上强加一个能够根据高度和横截面计算或至少估算体积的高级主体。许多教育工作者可能想把这样的公式强加给孩子。但这会引入一个与前保护主义儿童的智力体系格格不入的元素。我们的A几何学牢牢地属于儿童的体系。它甚至可能源自一位未能成功地为家庭建立秩序的父亲的模型。尽管我没有证据，但可以推测，保护的出现与儿童的俄狄浦斯危机有关，因为它赋予了这一模型显著性。我有更充分的理由猜测，像 geom 这样的东西之所以重要，是因为它具有用于构思乌龟的双向关系：它既与牢固存在的结构相关，比如孩子对权威人物的代表，也与重要数学思想的萌芽相关，比如“取消”的想法。
Despite its oversimplified, almost metaphorical status, the four-agent account of conservation captures an element of the paradox. A mathematical logician might like to impose on Aheight and Awidth a superior agent capable of calculating, or at least estimating, volume from height and cross-section. Many educators might like to impose such a formula on the child. But this would be introducing an element alien to the pre-conservationist child’s intellectual system. Our Ageom belongs firmly in the child’s system. It might even be derived from the model of a father not quite succeeding in imposing order on the family. It is possible to speculate, though I have no evidence, that the emergence of conservation is related to the child’s oedipal crisis through the salience it gives to this model. I feel on firmer ground in guessing that something like Ageom can become important because it so strongly has the two-sided relationship that was used to conceive the Turtle: It is related both to structures that are firmly in place, such as the child’s representation of authority figures, and to germs of important mathematical ideas, such as the idea of “cancellation.”
熟悉皮亚杰技术著作的读者会认识到“萌芽”这一概念是他分组中的原则之一。因此，他们可能会认为我们的模型与皮亚杰的模型没有太大区别。从根本上说，他们是正确的。但在赋予计算结构特殊作用方面引入了一个新元素：本书的主题是通过让儿童接触计算文化来利用这一特殊作用。如果且仅当这些结构正确，它们才可能极大地提高儿童以调动其概念潜力的方式表示现有结构的能力。
Readers who are familiar with Piaget’s technical writings will recognize this concept germ as one of the principles in his groupements. They may therefore see our model as not very different from Piaget’s. In a fundamental sense they would be right. But a new element is introduced in giving a special role to computational structures: The theme of this book has been the idea of exploiting this special role by giving children access to computational cultures. If, and only if, these have the right structure, they may greatly enhance children’s ability to represent the structures-in-place in ways that will mobilize their conceptual potential.
概括一下我们对皮亚杰理论的重新解读，有三点值得注意。首先，它提供了一种特定的心理学理论，其简洁性和解释力与该领域的其他理论相比极具竞争力。其次，它向我们展示了一种特定计算原理的力量，在这种情况下是纯程序理论，即可以封闭并以模块化方式使用的程序。第三，它具体化了我关于不同语言如何影响围绕它们成长的文化的论点。并非所有编程语言都体现了这种纯程序理论。如果没有，它们作为心理问题的隐喻的作用就会严重偏颇。人工智能和人工飞行之间的类比表明，无论这些现象看起来有多么不同，相同的原理都可能成为人工和自然现象的基础。升力动力学是飞行的基础，无论飞行者是血肉之躯还是金属。我们刚刚看到了一个可能对人类和人工智能都至关重要的原理：认识论模块化原理。关于实现智能的理想机器是模拟的还是数字的，以及大脑是模拟的还是数字的，存在许多争论。从我在此提出的理论的角度来看，这些争论都无关紧要。重要的问题不是大脑或计算机是否是离散的，而是知识是否可以模块化。
To recapitulate our reinterpretation of Piaget’s theory makes three points. First, it provides a specific psychological theory, highly competitive in its parsimony and explanatory power with others in the field. Second, it shows us the power of a specific computational principle, in this case the theory of pure procedures, that is, procedures that can be closed off and used in a modular way. Third, it concretizes my argument about how different languages can influence the cultures that can grow up around them. Not all programming languages embody this theory of pure procedures. When they do not, their role as metaphors for psychological issues is severely biased. The analogy between artificial intelligence and artificial flying made the point that the same principles could underlie the artificial and natural phenomena, however different these phenomena might appear. The dynamics of lift are fundamental to flight as such, whether the flyers are of flesh and blood or of metal. We have just seen a principle that may be fundamental both to human and artificial intelligence: the principle of epistemological modularity. There have been many arguments about whether the ideal machine for the achievement of intelligence would be analog or digital, and about whether the brain is analog or digital. From the point of view of the theory I am advancing here, these arguments are beside the point. The important question is not whether the brain or the computer is discrete but whether knowledge is modularizable.
对我来说，我们能够以这种方式使用计算隐喻作为新心理学理论的载体，这对知识理论的发展方向以及我们作为知识的生产者和载体的发展方向具有重要意义。这些领域并不是独立的。在前面的章节中，我们提出我们如何看待知识会影响我们如何看待自己。特别是，我们将知识分为不同种类的形象导致我们认为人们是根据他们的能力来划分的。这反过来又导致了我们文化的分裂。
For me, our ability to use computational metaphors in this way, as carriers for new psychological theories, has implications concerning where theories of knowledge are going and where we are going as producers and carriers of knowledge. These areas are not independent. In earlier chapters it was suggested that how we think about knowledge affects how we think about ourselves. In particular, our image of knowledge as divided up into different kinds leads us to a view of people as divided up according to what their aptitudes are. This in turn leads to a balkanization of our culture.
也许我需要澄清一下我为什么如此消极地谈论文化的分裂，却如此积极地谈论知识的模块化。当知识可以分解成“思维大小的碎片”时，它就更容易传播、更容易吸收、更容易构建。我们将知识分为科学和人文两个世界，这一事实决定了某些知识对于某些人来说是先验不可传播的。我们对交流的承诺不仅体现在我们对模块化的承诺上（模块化促进了交流），还体现在我们试图为物理和数学等领域寻找一种语言，而这些领域的本质是构造实体之间的交流。通过将牛顿定律重新表述为关于粒子（或“牛顿海龟”）如何相互交流的断言，我们为它提供了一个更容易被孩子或诗人抓住的把手。
Perhaps the fact that I have spoken so negatively about the balkanization of our culture and so positively about the modularization of knowledge requires some clarification. When knowledge can be broken up into “mind-size bites,” it is more communicable, more assimilable, more simply constructable. The fact that we divide knowledge up into scientific and humanistic worlds defines some knowledge as being a priori uncommunicable to certain kinds of people. Our commitment to communication is not only expressed through our commitment to modularization, which facilitates it, but through our attempt to find a language for such domains as physics and mathematics, which have as their essence communication between constructed entities. By restating Newton’s laws as assertions about how particles (or “Newtonian Turtles”) communicate with one another, we give it a handle that can be more easily grabbed by a child or by a poet.
再举一个例子，说明我们对知识的想象如何颠覆我们作为知识主体的自我意识。教育工作者有时会认为，知识的理想具有形式逻辑所定义的那种连贯性。但这些理想与大多数人体验自己的方式大相径庭。知识的主观体验更类似于竞争主体的混乱和争议，而不是p暗示q 的确定性和有序性。我们对自己的体验与我们对知识的理想化之间的差异会产生影响：它使我们感到恐惧，削弱了我们对自己能力的感觉，并导致我们采取适得其反的学习和思考策略。
Consider another example of how our images of knowledge can subvert our sense of ourselves as intellectual agents. Educators sometimes hold up an ideal of knowledge as having the kind of coherence defined by formal logic. But these ideals bear little resemblance to the way in which most people experience themselves. The subjective experience of knowledge is more similar to the chaos and controversy of competing agents than to the certitude and orderliness of p’s implying q’s. The discrepancy between our experience of ourselves and our idealizations of knowledge has an effect: It intimidates us, it lessens the sense of our own competence, and it leads us into counterproductive strategies for learning and thinking.
许多年纪较大的学生被吓得退学，成年人的情况更是如此。我们已经看到，尽管儿童认为自己是理论的创造者，但他们并没有因此受到尊重。这些矛盾因坚持一种没有人的思维方式符合的知识理想而加剧。许多儿童和大学生认为“我永远不可能成为数学家或科学家”，这反映出他们被引导相信数学家必须思考的方式与他们所知道的数学家思考方式之间存在差异。事实上，事实并非如此：他们自己的思维方式更像数学家，而不是逻辑理想。
Many older students have been intimidated to the point of dropping out, and what is true for adults is doubly true for children. We have already seen that despite their experience of themselves as theory builders, children are not respected as such. And these contradictions are compounded by holding out an ideal of knowledge to which no one’s thinking conforms. Many children and college students who decide “I can never be a mathematician or a scientist” are reflecting a discrepancy between the way they are led to believe the mathematician must think and the way they know they do. In fact the truth is otherwise: Their own thinking is much more like the mathematician’s than either is like the logical ideal.
我曾谈到强有力的思想对于理解世界的重要性。但是，如果我们每次学习新思想时都必须彻底重组我们的认知结构才能使用它，或者甚至必须确保没有引入不一致之处，那么我们几乎不可能学到新思想。尽管强有力的思想能够帮助我们组织思考某一类问题（如物理问题）的方式，但我们不必为了使用它们而重新组织自己。我们将我们的技能和启发式策略放入一种工具箱中——虽然它们的相互作用随着时间的推移会引起全球变化，但学习行为本身就是一个局部事件。
I have spoken of the importance of powerful ideas in grasping the world. But we could hardly ever learn a new idea if every time we did we had to totally reorganize our cognitive structures in order to use it or if we even had to ensure that no inconsistencies had been introduced. Although powerful ideas have the capacity to help us organize our way of thinking about a particular class of problems (such as physics problems), we don’t have to reorganize ourselves in order to use them. We put our skills and heuristic strategies into a kind of tool box—and while their interaction can, in the course of time, give rise to global changes, the act of learning is itself a local event.
学习的局部性体现在我对守恒定律习得的描述中。必要的代理在局部进入系统；它们的最高目标相互矛盾；最终协调它们的代理让它们保持原样。没有理由认为这种理论构建的“拼凑理论”只适用于描述儿童的学习。人工智能研究逐渐让我们更确定地认识到，根据我们为守恒定律问题勾勒出的模式，可以有效解决一系列问题：使用模块化代理，每个代理都有自己的简单之处，其中许多代理相互冲突。冲突受到调节和控制，而不是通过与原始代理一样简单的特殊代理的干预来“解决”。它们协调差异的方式并不涉及将系统强制为逻辑一致的模式。
The local nature of learning is seen in my description of the acquisition of conservation. The necessary agents entered the system locally; their top goals were in contradiction with each other; the agent that finally reconciles them leaves them in place. There is no reason why this “patchwork theory” of theory building should be considered appropriate only for describing the learning of children. Research in artificial intelligence is gradually giving us a surer sense of the range of problems that can be meaningfully solved on the pattern we have sketched for the conservation problem: with modular agents, each of them simple-minded in its own way, many of them in conflict with one other. The conflicts are regulated and kept in check rather than “resolved” through the intervention of special agents no less simple-minded than the original ones. Their way of reconciling differences does not involve forcing the system into a logically consistent mold.
这个过程让人想起修修补补；学习包括建立一套可以处理和操纵的材料和工具。也许最重要的是，这是一个利用你所拥有的东西的过程。我们在意识层面上都熟悉这个过程，例如，当我们从经验上解决问题时，尝试所有我们已知的以前解决过类似问题的方法。但在这里，我认为利用你所拥有的东西是更深层次、甚至是无意识的学习过程的简写。人类学家克劳德·列维-斯特劳斯2也曾以类似的方式谈到原始科学特有的理论构建。这是一门具体的科学，其中自然物体在其所有组合和重组中的关系为构建科学理论提供了概念词汇。在这里，我认为从最基本的意义上讲，作为学习者，我们都是拼贴者3。这引出了我们计算代理理论的第二种含义。如果第一个影响与我们对知识和学习的看法有关，那么第二个影响则可能与我们作为学习者的自我形象有关。如果拼贴是科学合理理论构建的典范，那么我们就可以开始更加尊重自己作为拼贴者的身份。当然，这与我们的中心主题——皮亚杰学习的重要性和力量——相联系。为了创造条件，将现在的非皮亚杰学习带入皮亚杰学习，我们必须能够真诚行事。我们必须感觉到，我们在这个过程中并没有让知识变质。
The process reminds one of tinkering; learning consists of building up a set of materials and tools that one can handle and manipulate. Perhaps most central of all, it is a process of working with what you’ve got. We’re all familiar with this process on the conscious level, for example, when we attack a problem empirically, trying out all the things that we have ever known to have worked on similar problems before. But here I suggest that working with what you’ve got is a shorthand for deeper, even unconscious learning processes. Anthropologist Claude Lévi-Strauss2 has spoken in similar terms of the kind of theory building that is characteristic of primitive science. This is a science of the concrete, where the relationships between natural objects in all their combinations and recombinations provide a conceptual vocabulary for building scientific theories. Here I am suggesting that in the most fundamental sense, we, as learners, are all bricoleurs.3 This leads us into the second kind of implication of our computational theory of agents. If the first implications had to do with impacts on our ideas about knowledge and learning, the second have to do with possible impacts on our images of ourselves as learners. If bricolage is a model for how scientifically legitimate theories are built, then we can begin to develop a greater respect for ourselves as bricoleurs. And of course this joins our central theme of the importance and power of Piagetian learning. In order to create the conditions for bringing what is now non-Piagetian learning to the Piagetian side, we have to be able to act in good faith. We have to feel that we are not denaturing knowledge in the process.
我用一个猜想来结束关于认知理论和人的这一章。之前我说过，我不会把皮亚杰描绘成一个阶段理论家。但思考皮亚杰的阶段理论确实提供了一个背景，让我们可以提出一个重要的观点，即计算文化对人类的可能影响。皮亚杰认为他的认知发展阶段是不变的，许多跨文化调查似乎证实了他信念的有效性。在一个又一个的社会里，孩子们似乎以同样的顺序发展认知能力。特别是，他的具体运算阶段（守恒通常属于这一阶段）比下一个也是最后一个阶段（形式运算阶段）早四年或更早开始。具体运算阶段的构建得到了以下观察的支持：通常，我们社会中的六七岁的孩子在许多领域取得了突破，而且似乎是同时取得的。他们能够使用数字、空间和时间单位；通过传递性进行推理；建立分类系统。但有些事情他们做不到。尤其是，在需要思考事情如何发展而不是现状如何的情况下，他们会陷入困境。让我们考虑以下我在介绍中预料到的例子。
I end this chapter on cognitive theory and people with a conjecture. Earlier I said that I would not present Piaget as a theorist of stages. But thinking about Piagetian stages does provide a context in which to make an important point about a possible impact of a computational culture on people. Piaget sees his stages of cognitive development as invariable, and numerous cross-cultural investigations have seemed to confirm the validity of his belief. In society after society, children seem to develop cognitive capacities in the same order. In particular, his stage of concrete operations, to which the conservations typically belong, begins four or more years earlier than the next and final stage, the stage of formal operations. The construct of a stage of concrete operations is supported by the observation that, typically, children in our society at six or seven make a breakthrough in many realms, and seemingly all at once. They are able to use units of numbers, space, and time; to reason by transitivity; to build up classificatory systems. But there are things they cannot do. In particular, they flounder in situations that call for thinking not about how things are but about all the ways they could be. Let us consider the following example, which I anticipated in the introduction.
给孩子一组不同颜色的珠子，比如绿色、红色、蓝色和黑色，并要求他们构造所有可能的颜色组合：绿蓝、绿红、绿黑，然后是三色组合等等。就像孩子直到 7 岁才掌握守恒定律一样，世界各地的孩子在 11 岁或 12 岁之前都无法完成这样的组合任务。事实上，许多“聪明”到可以过正常生活的成年人从未获得这种能力。
A child is given a collection of beads of different colors, say green, red, blue, and black, and is asked to construct all the possible pairs of colors: green-blue, green-red, green-black, and then the triplets and so on. Just as children do not acquire conservation until their seventh year, children around the world are unable to carry out such combinatorial tasks before their eleventh or twelfth year. Indeed, many adults who are “intelligent” enough to live normal lives never acquire this ability.
守恒中所谓的“具体”操作与组合任务中所谓的“形式”操作之间的区别究竟是什么？皮亚杰给它们起的名字和经验数据表明了深刻而本质的区别。但从这里提出的思想的角度看这个问题，给人的印象却大不相同。
What is the nature of the difference between the so-called “concrete” operations involved in conservation and the so-called “formal” operations involved in the combinatorial task? The names given them by Piaget and the empirical data suggest a deep and essential difference. But looking at the problem through the prism of the ideas developed here gives a much different impression.
从计算的角度来看，组合任务最突出的要素与程序的理念有关——系统性和调试。成功的解决方案包括遵循以下一些程序：
From a computational point of view, the most salient ingredients of the combinatorial task are related to the idea of procedure—systematicity and debugging. A successful solution consists of following some such procedure as:
1. 将珠子按照颜色分开。
1. Separate the beads into colors.
2. 选择颜色A作为颜色 1。
2. Choose a color A as color 1.
3. 组成所有可以用颜色 1 形成的对。
3. Form all the pairs that can be formed with color 1.
4. 选择颜色 2。
4. Choose color 2.
5. 组成所有可以用颜色 2 形成的对。
5. Form all the pairs that can be formed with color 2.
6. 对每种颜色都执行此操作。
6. Do this for each color.
7.返回并删除重复项。
7. Go back and remove the duplicates.
因此，真正涉及的是编写和执行程序，包括最重要的调试步骤。这一观察表明了儿童如此晚才获得这种能力的原因：当代文化为这种类型的系统程序元素提供了相对较少的机会。我并不是说没有这样的机会。有些机会是可以遇到的；例如，在游戏中，孩子可以创造自己的“组合微世界”。但在这方面为孩子提供的机会、激励和帮助比在数字等领域少得多。在我们的文化中，数字表现丰富，而系统程序表现不佳。4
So what is really involved is writing and executing a program including the all-important debugging step. This observation suggests a reason for the fact that children acquire this ability so late: Contemporary culture provides relatively little opportunity for bricolage with the elements of systematic procedures of this type. I do not mean to say that there are no such opportunities. Some are encountered; for example, in games where a child can create his own “combinatorial microworlds.” But the opportunities, the incentives, and the help offered the child in this area are very significantly less than in such areas as number. In our culture number is richly represented, and systematic procedure is poorly represented.4
我认为没有理由怀疑这种差异可以解释获得数字守恒和组合能力的年龄之间五年或五年以上的差距。
I see no reason to doubt that this difference could account for a gap of five years or more between the ages at which conservation of number and combinatorial abilities are acquired.
研究此类假设的标准方法是比较不同文化背景下的儿童。当然，这种方法是针对皮亚杰阶段进行的。人类学家已经能够区分处于各个发展阶段的儿童，并在各大洲的 100 多个不同社会中要求他们倒液体和分类珠子。在所有情况下，如果守恒定律和组合技能都具备，那么数字守恒定律在年龄小于那些具备组合技能的儿童身上就表现出来。然而，这一观察结果毫不怀疑我的假设。在前计算机社会中，数字知识比编程知识更丰富，这很可能是普遍事实。不难为这种认知社会普遍现象找到合理的解释。但在未来的计算机丰富的文化中，情况可能会有所不同。如果计算机和编程成为儿童日常生活的一部分，守恒定律和组合之间的差距肯定会缩小，而且可以想象得到逆转：儿童可能先学会系统化，然后才学会定量化！
The standard methodology for investigating such hypotheses as this is to compare children in different cultures. This has, of course, been done for the Piagetian stages. Children at all the levels of development anthropologists have been able to distinguish, and in over a hundred different societies from all the continents, have been asked to pour liquids and sort beads. In all cases, if conservation and combinatorial skills came at all, conservation of numbers was evidenced by children five or more years younger than those evidencing combinatorial skills. Yet this observation casts no doubt on my hypothesis. It may well be universally true of precomputer societies that numerical knowledge would be more richly represented than programming knowledge. It is not hard to invent plausible explanations of such a cognitive-social universal. But things may be different in the computer-rich cultures of the future. If computers and programming become a part of the daily life of children, the conservation-combinatorial gap will surely close and could conceivably be reversed: Children may learn to be systematic before they learn to be quantitative!
脚注
Footnote
接下来从计算的角度对保护进行了高度系统化和简化的概述，说明了如何用马文·明斯基和作者正在开发的“心智社会”理论来解释这一现象，我们将在即将出版的书中讨论这一问题。
I The computational perspective on conservation that follows is a highly schematized and simplified overview of how this phenomenon would be explained by a theory, “The Society of Mind,” being developed by Marvin Minsky and the author and to be discussed in our forthcoming book.
第八章
CHAPTER 8
学习型社会的形象
Images of the Learning Society
我所提出的愿景是一种特殊的计算机文化，一种数学文化，即一种不仅能帮助我们学习，还能帮助我们学会学习的文化。我已经说明了这种文化如何通过允许与知识建立更个人化、更少疏离的关系来使学习人性化，并举了一些例子来说明它如何改善与学习过程中遇到的其他人的关系：同学和老师。但我只是对这种学习可能发生的社会背景做了一些简要的评论。现在是时候面对（尽管我无法回答）许多读者心中必定存在的问题了：这种环境是学校吗？
THE VISION I HAVE PRESENTED IS OF A PARTICULAR COMPUTER CULture, a mathetic one, that is, one that helps us not only to learn but to learn about learning. I have shown how this culture can humanize learning by permitting more personal, less alienating relationships with knowledge and have given some examples of how it can improve relationships with other people encountered in the learning process: fellow students and teachers. But I have made only passing remarks about the social context in which this learning might take place. It is time to face (though I cannot answer) a question that must be in many readers’ minds: Will this context be school?
学校可能有一天会不复存在，这一说法引起了许多人的强烈反应。要清晰地思考一个没有学校的世界，会遇到许多障碍。有些障碍非常个人化。我们大多数人一生中在学校度过的时间比我们愿意想象的要多。例如，我已年过五十，但我在学校毕业后的年数几乎赶不上我在学前和在学校的年数。没有我们自己的生活经验，没有学校的世界这一概念是非常不协调的。其他障碍则更具概念性。我们不能消极地定义这样一个世界，即简单地取消学校，不放任何东西来代替它。这样做会留下一个思想真空，大脑必须以某种方式填补这个真空，通常是模糊但可怕的形象，比如孩子们“放荡不羁”、“吸毒”或“让父母的生活无法维持”。认真思考一个没有学校的世界需要精心设计儿童将参与的非学校活动模型。
The suggestion that there might come a day when schools no longer exist elicits strong response from many people. There are many obstacles to thinking clearly about a world without schools. Some are highly personal. Most of us spent a larger fraction of our lives going to school than we care to think about. For example, I am over fifty and yet the number of my postschool years has barely caught up with my preschool and school years. The concept of a world without school is highly dissonant without experiences of our own lives. Other obstacles are more conceptual. One cannot define such a world negatively, that is by simply removing school and putting nothing in its place. Doing so leaves a thought vacuum that the mind has to fill one way or another, often with vague but scary images of children “running wild,” “drugging themselves,” or “making life impossible for their parents.” Thinking seriously about a world without schools calls for elaborated models of the nonschool activities in which children would engage.
对我来说，收集这样的模型已经成为思考儿童未来的一个重要部分。最近我在巴西度过了一个夏天，发现了一个绝佳的模型。例如，里约热内卢著名的狂欢节的核心是长达十二小时的歌舞和街头戏剧游行。一队又一队演员表演自己的作品。通常，作品是通过音乐和舞蹈演绎历史事件或民间故事。歌词、舞蹈编排、服装都是新颖的。技术成就水平是专业的，效果令人叹为观止。虽然参考可能是神话，但游行充满了当代的政治意义。
For me, collecting such models has become an important part of thinking about the future of children. I recently found an excellent model during a summer spent in Brazil. For example, at the core of the famous carnival in Rio de Janeiro is a twelve-hour-long procession of song, dance, and street theater. One troop of players after another presents its piece. Usually the piece is a dramatization through music and dance of a historical event or folktale. The lyrics, the choreography, the costumes are new and original. The level of technical achievement is professional, the effect breathtaking. Although the reference may be mythological, the processions are charged with contemporary political meaning.
游行并非自发的。准备游行和表演游行都是巴西人生活的重要组成部分。每个团体都在自己的学习环境中单独准备，并进行竞争，这个学习环境被称为桑巴舞学校。这些不是我们所知的学校；它们是社交俱乐部，会员人数从几百人到几千人不等。每个俱乐部都拥有一栋建筑，一个跳舞和聚会的地方。桑巴舞学校的成员大多数周末晚上都会去那里跳舞、喝酒、会见朋友。
The processions are not spontaneous. Preparing them as well as performing in them are important parts of Brazilian life. Each group prepares separately—and competitively—in its own learning environment, which is called a samba school. These are not schools as we know them; they are social clubs with memberships that may range from a few hundred to many thousands. Each club owns a building, a place for dancing and getting together. Members of a samba school go there most weekend evenings to dance, to drink, to meet their friends.
每年，每个桑巴舞学校都会为下一届狂欢节选择主题，选出明星，反复编写歌词，编排和练习舞蹈。学校成员的年龄从儿童到祖父母不等，水平从新手到专业人士不等。但他们一起跳舞，跳舞时每个人都在学习和教学。甚至明星们也在那里学习他们最难的部分。
During the year each samba school chooses its theme for the next carnival, the stars are selected, the lyrics are written and rewritten, the dance is choreographed and practiced. Members of the school range in age from children to grandparents and in ability from novice to professional. But they dance together and as they dance everyone is learning and teaching as well as dancing. Even the stars are there to learn their difficult parts.
每个美国迪斯科舞厅都是学习和跳舞的地方。但桑巴舞学校却大不相同。那里的社会凝聚力更强，有团体归属感，有共同目标感。虽然大部分教学是在自然环境中进行的，但都是经过深思熟虑的。例如，一位专业舞者会把一群孩子聚集在一起。五到二十分钟后，一个特定的学习小组就出现了。他们的学习是经过深思熟虑和专注的。然后他们就融入了人群中。
Every American disco is a place for learning as well as for dancing. But the samba schools are very different. There is a greater social cohesion, a sense of belonging to a group, and a sense of common purpose. Much of the teaching, although it takes place in a natural environment, is deliberate. For example, an expert dancer gathers a group of children around. For five or for twenty minutes a specific learning group comes into existence. Its learning is deliberate and focused. Then it dissolves into the crowd.
在本书中，我们探讨了如何在类似巴西桑巴舞学校的环境中学习数学，在真实的、具有社会凝聚力的环境中，专家和新手都在学习。桑巴舞学校虽然不能“输出”到外来文化，但它代表了学习环境应该和可以拥有的一组属性。学习与现实并不分离。桑巴舞学校有一个目的，学习就是为了这个目的而融入学校的。新手与专家并不分离，专家也在学习。
In this book we have considered how mathematics might be learned in settings that resemble the Brazilian samba school, in settings that are real, socially cohesive, and where experts and novices are all learning. The samba school, although not “exportable” to an alien culture, represents a set of attributes a learning environment should and could have. Learning is not separate from reality. The samba school has a purpose, and learning is integrated in the school for this purpose. Novice is not separated from expert, and the experts are also learning.
LOGO环境在某些方面与桑巴舞学校相似，但在其他方面又有所不同。最相似之处在于，在 LOGO 环境中，数学是一项真正的活动，新手和专家都可以参与其中。这项活动如此多样化，如此富有发现性，以至于即使在编程的第一天，学生也可能做一些对老师来说很新奇和令人兴奋的事情。约翰·杜威表达了对早期社会的怀念，在早期社会中，孩子们通过真正的参与和嬉戏的模仿成为猎人。今天，我们学校的学习并没有太多的参与性——做算术也不是对成人生活中令人兴奋、可识别活动的模仿。但编写计算机图形或音乐程序和驾驶模拟宇宙飞船确实与成年人的真实活动有很多相似之处，甚至与那种可以成为雄心勃勃的孩子的英雄和榜样的成年人非常相似。
LOGO environments are like samba schools in some ways, unlike them in other ways. The deepest resemblance comes from the fact that in them mathematics is a real activity that can be shared by novices and experts. The activity is so varied, so discovery-rich, that even in the first day of programming, the student may do something that is new and exciting to the teacher. John Dewey expressed a nostalgia for earlier societies where the child becomes a hunter by real participation and by playful imitation. Learning in our schools today is not significantly participatory—and doing sums is not an imitation of an exciting, recognizable activity of adult life. But writing programs for computer graphics or music and flying a simulated spaceship do share very much with the real activities of adults, even with the kind of adult who could be a hero and a role model for an ambitious child.
LOGO环境在人际关系质量方面也与桑巴舞学校相似。虽然教师通常会在场，但他们的干预更类似于桑巴舞学校的专业舞者，而不是手持教案和既定课程的传统教师。LOGO教师会回答问题，在被问到时提供帮助，有时会坐在学生旁边说：“让我给你看点东西。”所展示的内容不受既定教学大纲的限制。有时是学生可以用于即时项目的东西。有时是教师最近学到的，并认为学生会喜欢的东西。有时教师只是自发地行动，就像人们在所有非结构化的社交场合中对自己所做的事情感到兴奋时所做的那样。LOGO环境与桑巴舞学校的另一个相似之处在于，思想的流动甚至指令的流动都不是单向的。该环境旨在促进比当今学校中与任何数学相关的内容更丰富、更深入的互动。孩子们可以创建程序，产生令人愉悦的图形、有趣的图片、音效、音乐和计算机笑话。他们开始以数学的方式互动，因为他们的数学工作成果属于他们，属于现实生活。乐趣的一部分是分享、在墙上张贴图形、修改和试验彼此的作品，并将“新”产品带回给最初的发明者。尽管在电脑前工作通常是私密的，但它增加了孩子们互动的欲望。这些孩子希望与从事类似活动的其他人聚在一起，因为他们有很多话要说。他们要说的话不仅限于谈论他们的产品：LOGO旨在让人们轻松讲述制造产品的过程。
LOGO environments also resemble samba schools in the quality of their human relationships. Although teachers are usually present, their interventions are more similar to those of the expert dancers in the samba school than those of the traditional teacher armed with lesson plans and a set curriculum. The LOGO teacher will answer questions, provide help if asked, and sometimes sit down next to a student and say: “Let me show you something.” What is shown is not dictated by a set syllabus. Sometimes it is something the student can use for an immediate project. Sometimes it is something that the teacher has recently learned and thinks the student would enjoy. Sometimes the teacher is simply acting spontaneously as people do in all unstructured social situations when they are excited about what they are doing. The LOGO environment is like the samba school also in the fact that the flow of ideas and even of instructions is not a one-way street. The environment is designed to foster richer and deeper interactions than are commonly seen in schools today in connection with anything mathematical. Children create programs that produce pleasing graphics, funny pictures, sound effects, music, and computer jokes. They start interacting mathematically because the product of their mathematical work belongs to them and belongs to real life. Part of the fun is sharing, posting graphics on the walls, modifying and experimenting with each other’s work, and bringing the “new” products back to the original inventors. Although the work at the computer is usually private it increases the children’s desire for interaction. These children want to get together with others engaged in similar activities because they have a lot to talk about. And what they have to say to one another is not limited to talking about their products: LOGO is designed to make it easy to tell about the process of making them.
通过构建LOGO，让结构化思维变成强大的思维，我们传达了一种认知风格，其中的一个方面就是促进思考过程的讨论。LOGO对调试的重视也朝着同一个方向发展。学生的 bug 成为谈话的主题；因此，他们会发展出一种清晰而专注的语言，以便在需要时寻求帮助。当需要帮助时，帮助者不一定非得是经过专门培训的专业人士才能提供帮助。通过这种方式，LOGO文化丰富和促进了所有参与者之间的互动，并为更清晰、更有效、更诚实的教学关系提供了机会。这是朝着学习者和教师之间的界限逐渐模糊的方向迈出的一步。
By building LOGO in such a way that structured thinking becomes powerful thinking, we convey a cognitive style, one aspect of which is to facilitate talking about the process of thinking. LOGO’s emphasis on debugging goes in the same direction. Students’ bugs become topics of conversation; as a result they develop an articulate and focused language to use in asking for help when it is needed. And when the need for help can be articulated clearly, the helper does not necessarily have to be a specially trained professional in order to give it. In this way the LOGO culture enriches and facilitates the interaction between all participants and offers opportunities for more articulate, effective, and honest teaching relationships. It is a step toward a situation in which the line between learners and teachers can fade.
尽管有这些相似之处，但LOGO环境并不是桑巴舞学校。两者的区别非常根本。表面上看，这些区别体现在教师都是专业人士，即使不行使权力，他们也掌控着一切。学生是临时群体，很少能待足够长的时间来实现LOGO的长期目标。归根结底，区别在于这两个实体与周围文化的关系。桑巴舞学校与流行文化有着丰富的联系。在那里学习的知识与这种文化是连续的。LOGO环境是人工维护的绿洲，人们在这里接触到与周围文化主流分离的知识（数学和数学），事实上，这种知识甚至与周围文化所表达的价值观有些对立。当我问自己这种情况是否可以改变时，我会提醒自己这个问题的社会性质，记住桑巴舞学校不是由研究人员设计的，不是由拨款资助的，也不是由政府行动实施的。它不是被制造出来的。它是自然发生的。这对于任何可能从数学计算机文化中诞生的新的、成功的学习协会形式来说也一定是正确的。强大的新社会形式必须扎根于文化，而不是官僚的产物。
Despite these similarities, LOGO environments are not samba schools. The differences are quite fundamental. They are reflected superficially in the fact that the teachers are professionals and are in charge even when they refrain from exerting authority. The students are a transitory population and seldom stay long enough to make LOGO’s long-term goals their own. Ultimately the difference has to do with how the two entities are related to the surrounding culture. The samba school has rich connections with a popular culture. The knowledge being learned there is continuous with that culture. The LOGO environments are artificially maintained oases where people encounter knowledge (mathematical and mathetic) that has been separated from the mainstream of the surrounding culture, indeed which is even in some opposition to values expressed in that surrounding culture. When I ask myself whether this can change, I remind myself of the social nature of the question by remembering that the samba school was not designed by researchers, funded by grants, nor implemented by government action. It was not made. It happened. This must be true too of any new successful forms of associations for learning that might emerge out of the mathetic computer culture. Powerful new social forms must have their roots in the culture, not be the creatures of bureaucrats.
因此，我们又回到了教育者成为人类学家的必要性。教育创新者必须意识到，要想取得成功，他们必须对周围文化中发生的事情保持敏感，并利用动态的文化趋势作为媒介来进行教育干预。
Thus we are brought back to seeing the necessity for the educator to be an anthropologist. Educational innovators must be aware that in order to be successful they must be sensitive to what is happening in the surrounding culture and use dynamic cultural trends as a medium to carry their educational interventions.
当今文化以无处不在的计算机技术为标志，这种说法已成常态。一段时间以来，这种说法一直是正确的。但近年来，出现了一些新情况。在过去两年中，超过 20 万台个人计算机进入了美国人的生活，其中一些最初是为了工作而不是娱乐或教育目的而购买的。然而，对于教育工作者兼人类学家来说，重要的是，它们作为人们看到的物品而存在，并开始接受它们作为日常生活的一部分。在技术大规模渗透的同时，一场与教育政治密切相关的社会运动正在展开。人们对传统教育的失望情绪日益加深。有些人通过极端行动来表达这种情绪，他们实际上让孩子退学，选择在家接受教育。对大多数人来说，他们只是感到学校已经不再能起到作用了。我相信这两种趋势可以结合在一起，这对孩子、父母和学习都有好处。这是通过构建教育上强大的计算环境来实现的，这些环境将为传统教室和传统教学提供替代方案。我并没有提出LOGO环境作为我的建议。它们太原始了，太受 20 世纪 70 年代技术的限制了。我希望它们能发挥模型的作用。现在读者一定已经预料到，我将说一个思考对象，它将为构建未来教育的本质上是社会的过程做出贡献。
It has become commonplace to say that today’s culture is marked by a ubiquitous computer technology. This has been true for some time. But in recent years, there is something new. In the past two years, over 200,000 personal computers have entered the lives of Americans, some of them originally bought for business rather than recreational or educational purposes. What is important to the educator-as-anthropologist, however, is that they exist as objects that people see, and start to accept, as part of the reality of everyday life. And at the same time that this massive penetration of the technology is taking place, there is a social movement afoot with great relevance for the politics of education. There is an increasing disillusion with traditional education. Some people express this by extreme action, actually withdrawing their children from schools and choosing to educate them at home. For most, there is simply the gnawing sense that schools simply aren’t doing the job anymore. I believe that these two trends can come together in a way that would be good for children, for parents, and for learning. This is through the construction of educationally powerful computational environments that will provide alternatives to traditional classrooms and traditional instruction. I do not present LOGO environments as my proposal for this. They are too primitive, too limited by the technology of the 1970s. The role I hope they fill is that of a model. By now the reader must anticipate that I shall say an object-to-think-with, that will contribute to the essentially social process of constructing the education of the future.
LOGO环境不是桑巴舞学校，但它们有助于想象“数学桑巴舞学校”的情景。直到最近，这种事情才成为可能。计算机通过提供数学丰富的活动将其带入了可能领域，这些活动原则上可以真正吸引新手和专家、年轻人和老年人。我毫不怀疑，在未来几年内，我们将看到一些值得称为“计算桑巴舞学校”的计算环境的形成。参与计算机爱好者俱乐部和经营计算机“临时中心”的人已经在这方面进行了尝试。
LOGO environments are not samba schools, but they are useful for imagining what it would be like to have a “samba school for mathematics.” Such a thing was simply not conceivable until very recently. The computer brings it into the realm of the possible by providing mathematically rich activities which could, in principle, be truly engaging for novice and expert, young and old. I have no doubt that in the next few years we shall see the formation of some computational environments that deserve to be called “samba schools for computation.” There have already been attempts in this direction by people engaged in computer hobbyist clubs and in running computer “drop-in centers.”
在大多数情况下，尽管这些实验很有趣也很令人兴奋，但由于太过原始，它们还是以失败告终。他们的计算机根本无法满足最吸引人、最可共享的活动所需的能力。他们对如何将计算思维融入日常生活的设想还不够成熟。但还会有更多尝试，而且会越来越多。最终，在某个地方，所有的部分都会组合在一起，并“流行起来”。我们可以对此充满信心，因为这些尝试不会是孤立的实验，不会由那些可能耗尽资金或只是幻想破灭而放弃的研究人员进行。它们将成为人们对个人计算、对自己的孩子和对教育感兴趣的社会运动的体现。
In most cases, although the experiments have been interesting and exciting, they have failed to make it because they were too primitive. Their computers simply did not have the power needed for the most engaging and shareable kinds of activities. Their visions of how to integrate computational thinking into everyday life was insufficiently developed. But there will be more tries, and more and more. And eventually, somewhere, all the pieces will come together and it will “catch.” One can be confident of this because such attempts will not be isolated experiments operated by researchers who may run out of funds or simply become disillusioned and quit. They will be manifestations of a social movement of people interested in personal computation, interested in their own children, and interested in education.
桑巴舞学校作为教育中心的形象存在问题。我相信计算机桑巴舞学校会在某个地方流行起来。但第一个学校几乎肯定会出现在一个特定类型的社区，可能是中等收入工程师密度高的社区。这将使计算机桑巴舞学校扎下“文化根基”，但当然，它也会在桑巴舞学校的文化上留下印记。对于对教育感兴趣的人来说，追溯这些努力的历程很重要：它们将如何影响学龄参与者的智力发展？我们会看到皮亚杰阶段的逆转吗？他们会面临退出传统学校的压力吗？当地学校将如何努力适应新的压力？但作为一个教育乌托邦主义者，我想要别的东西。我想知道在那些还没有丰富的技术爱好者土壤的社区里，什么样的计算机文化可以成长。我想知道，我想帮助实现它。
There are problems with the image of samba schools as the locus of education. I am sure that a computational samba school will catch on somewhere. But the first one will almost certainly happen in a community of a particular kind, probably one with a high density of middle-income engineers. This will allow the computer samba school to put down “cultural roots,” but it will, of course, also leave its mark on the culture of the samba school. For people interested in education in general, it will be important to trace the life histories of these efforts: How will they affect the intellectual development of their school-age participants? Will we see reversals of Piagetian stages? Will they develop pressures to withdraw from traditional schools? How will local schools try to adapt to the new pressure on them? But as an educational utopian I want something else. I want to know what kind of computer culture can grow in communities where there is not already a rich technophilic soil. I want to know and I want to help make it happen.
让我再说一遍，潜在的障碍不是经济上的，也不是说计算机不会成为人们日常生活中的物品。它们最终会成为的。它们已经进入大多数工作场所，并最终将进入大多数家庭，就像现在的电视机一样，而且在许多情况下，最初是出于同样的原因。流行计算机文化发展的障碍是文化上的，例如，当今机器中嵌入的计算机文化与它们将进入的家庭文化不匹配。如果问题出在文化上，那么补救措施也必须是文化上的。
Let me say once more, the potential obstacle is not economic and it is not that computers are not going to be objects in people’s everyday lives. They eventually will. They are already entering most workplaces and will eventually go into most homes just as TV sets now do, and in many cases initially for the same reasons. The obstacle to the growth of popular computer cultures is cultural, for example, the mismatch between the computer culture embedded in the machines of today and the cultures of the homes they will go into. And if the problem is cultural the remedy must be cultural.
研究挑战显而易见。我们需要推进计算机与文化融合的艺术，以便它们能够将当代社会中共存的支离破碎的亚文化统一起来，但希望不会使之同质化。例如，必须弥合技术科学文化与人文文化之间的鸿沟。我认为，搭建这座桥梁的关键是学习如何以计算形式重塑强大的思想，这些思想对诗人和工程师同样重要。
The research challenge is clear. We need to advance the art of meshing computers with cultures so that they can serve to unite, hopefully without homogenizing, the fragmented subcultures that coexist counterproductively in contemporary society. For example, the gulf must be bridged between the technical-scientific and humanistic cultures. And I think that the key to constructing this bridge will be learning how to recast powerful ideas in computational form, ideas that are as important to the poet as to the engineer.
在我看来，计算机充当着一个过渡对象，用于调解人与人之间最终的关系。有些人对移动身体有着敏锐的感觉，有些人对数学却忘记了数学知识的感觉运动根源。海龟建立了一座桥梁。它充当着一种共同的媒介，可以重塑身体几何学和形式几何学的共同元素。将杂耍重新塑造为结构化编程，可以在那些对身体技能有着敏锐数学感觉的人和那些知道如何组织历史论文写作任务的人之间架起一座桥梁。
In my vision the computer acts as a transitional object to mediate relationships that are ultimately between person and person. There are mathophobes with a fine sense of moving their bodies, and there are mathophiles who have forgotten the sensory motor roots of their mathematical knowledge. The Turtle establishes a bridge. It serves as a common medium in which can be recast the shared elements of body geometry and formal geometry. Recasting juggling as structured programming can build a bridge between those who have a fine mathetic sense of physical skills and those who know how to go about organizing the task of writing an essay on history.
如果从结果来看，玩杂耍和写文章似乎没有什么共同之处。但学习这两种技能的过程却有很多共同之处。通过创造一个强调过程的智力环境，我们为拥有不同技能和兴趣的人提供了可以谈论的话题。通过开发用于谈论过程的富有表现力的语言，并通过用这些新语言重塑旧知识，我们希望能够消除学科之间的障碍。在学校里，数学就是数学，历史就是历史，玩杂耍超出了智力的范围。时间将告诉我们学校是否能够适应。更重要的是理解将知识重塑为新形式。
Juggling and writing an essay seem to have little in common if one looks at the product. But the processes of learning both skills have much in common. By creating an intellectual environment in which the emphasis is on process, we give people with different skills and interests something to talk about. By developing expressive languages for talking about process and by recasting old knowledge in these new languages, we can hope to make transparent the barriers separating disciplines. In the schools, math is math and history is history and juggling is outside the intellectual pale. Time will tell whether schools can adapt themselves. What is more important is understanding the recasting of knowledge into new forms.
在本书中，我们看到了新技术与主题重塑之间的复杂相互作用。当我们讨论使用计算机来促进学习牛顿运动定律时，我们并没有试图将方程式“计算机化”，就像经典教科书中的方程式一样。我们开发了一个用于思考运动的新概念框架。例如，海龟的概念使我们能够制定牛顿物理学的定性组成部分。由此产生的重新概念化即使没有计算机也是有效的；它与计算机的关系根本不是还原论的。但它能够以其他物理学概念化无法做到的方式利用计算机，从而获得数学能力。因此，整个过程涉及新技术与新物理方法之间的辩证互动。通过查看我收集的思考教育的优秀模型中的另一个项目，可以非常清楚地看到这些相互作用的逻辑。
In this book we have seen complex interactions between new technologies and the recasting of the subject matters. When we discussed the use of the computer to facilitate learning Newton’s laws of motion, we did not attempt to “computerize” the equations as they are found in a classical textbook. We developed a new conceptual framework for thinking about motion. For example, the concept of Turtle enabled us to formulate a qualitative component of Newtonian physics. The resulting reconceptualizing would be valid without a computer; its relation to the computer is not at all reductionist. But it is able to take advantage of the computer in ways in which other conceptualizations of physics could not, and thus gain in mathetic power. Thus, the whole process involves a dialectical interaction between new technologies and new ways of doing physics. The logic of these interactions is seen very clearly by looking at another item from my collection of good models for thinking about education.
二十年前，人们认为平行滑雪是一项需要经过多年训练和练习才能掌握的技能。如今，人们通常只需一个滑雪季就能掌握这项技能。促成这一变化的一些因素符合传统的教育创新模式。
Twenty years ago, parallel skiing was thought to be a skill attainable only after many years of training and practice. Today, it is routinely achieved during the course of a single skiing season. Some of the factors that contributed to this change are of a kind that fit into the traditional paradigms for educational innovation.
例如，许多滑雪学校采用一种新的教学方法（渐进长度法 - GLM），即首先学习使用短滑雪板滑雪，然后逐渐进步到更长的滑雪板。但发生了一些更根本的事情。从某种意义上说，今天新滑雪者如此轻松学习的东西与他们的父母认为如此困难的东西不同。父母的所有目标都由孩子们实现：滑雪者以平行的滑雪板迅速越过山峰，避开障碍物并穿越障碍门。但他们为实现这些结果而采取的动作却截然不同。
For example, many ski schools use a new pedagogical technique (the graduated length method—GLM) in which one first learns to ski using short skis and then gradually progresses to longer ones. But something more fundamental happened. In a certain sense what new skiers learn today so easily is not the same thing that their parents found so hard. All the goals of the parents are achieved by the children: The skiers move swiftly over the mountain with their skis parallel, avoiding obstacles and negotiating slalom gates. But the movements they make in order to produce these results are quite different.
当父母学习滑雪时，度假滑雪者和奥运冠军都使用基于预备反向旋转的转弯技巧，这被认为是平行转弯的必要条件。更直接的动作可以产生更有效的转弯这一认识是一项根本性的发现，它迅速改变了滑雪，无论是度假滑雪者还是冠军。对于新手来说，新技术意味着更快的学习，对于冠军来说，它意味着更有效的动作，对于时尚滑雪者来说，它意味着更多优雅动作的机会。因此，变化的核心是对滑雪本身的重新概念化，而不仅仅是教学法或技术的变化。但为了有一个完整的画面，我们还必须认识到内容、教学法和技术之间的辩证互动。因为随着滑雪动作的变化，滑雪板和靴子也在发生变化。新的塑料使靴子变得更轻更硬，而滑雪板可以变得更灵活或更不灵活。这些变化的方向与新的滑雪技术如此协同，以至于许多滑雪教练和滑雪作家将滑雪的变化归因于这项技术。同样，使用短滑雪板进行教学恰好与新技术高度适应，以至于许多人将滑雪革命总结为“转向GLM ”。
When the parents learned to ski, both vacation skiers and Olympic champions used turning techniques based on a preparatory counterrotation, thought to be necessary for parallel turns. The realization that more direct movements could produce a more effective turn was a fundamental discovery, and it rapidly transformed skiing, both for the vacation skier and the champion. For the novice the new techniques meant more rapid learning, for the champion it meant more efficient movements, for the fashionable skier it meant more opportunities for elegant movements. Thus, at the heart of the change is a reconceptualization of skiing itself, not a mere change in pedagogy or technology. But in order to have a complete picture, we must also recognize a dialectical interaction between the content, the pedagogy, and the technology. For as ski movements were changing, skis and boots were changing too. New plastics allowed boots to become lighter and more rigid, and skis could be made more or less flexible. The direction of these changes was so synergistic with the new ski techniques that many ski instructors and ski writers attributed the change in skiing to the technology. Similarly, the use of short skis for instruction happened to be so highly adaptable to the new technology that many people sum up the ski revolution as the “move to GLM.”
我喜欢思考“滑雪革命”，因为它可以帮助我思考我们在“计算机革命”历史中所处的非常复杂的节点。今天，我们听到很多关于“计算机即将到来”的讨论，也听到很多关于计算机将如何改变教育的讨论。大部分讨论分为两类，一类显然是“革命性的”，另一类是“改革性的”。对于许多革命者来说，计算机的出现本身将带来重大变化：家庭中的教学机器和计算机网络将使学校（正如我们所知）过时；重新概念化物理学是他们最不想做的事情。对于改革者来说，计算机不会废除学校，而是会为学校服务。计算机被视为一种引擎，可以利用现有结构，以局部和渐进的方式解决当今学校面临的问题。改革者并不比革命者更倾向于从重新概念化学科领域的角度来思考。
I like to think about the “ski revolution” because it helps me to think about the very complex junction we are at in the history of the “computer revolution.” Today we hear a lot of talk about how “computers are coming” and a lot of talk about how they will change education. Most of the talk falls into two categories, one apparently “revolutionary” and the other “reformist.” For many revolutionaries, the presence of the computer will in itself produce momentous change: Teaching machines in the homes and computer networks will make school (as we know it) obsolete; reconceptualizations of physics are the furthest things from their minds. For the reformists, the computer will not abolish schools but will serve them. The computer is seen as an engine that can be harnessed to existing structures in order to solve, in local and incremental measures, the problems that face schools as they exist today. The reformist is no more inclined than the revolutionary to think in terms of reconceptualizing subject domains.
我们的理念，无论是隐含的还是明确的，都试图避免两个常见的陷阱：对技术必然性的坚持和对渐进式变革策略的坚持。技术本身不会引导我们朝着我认为的任何方向前进，无论是在教育上还是在社会上。教育界被动的代价将是教育平庸和社会僵化。而尝试渐进式变革甚至不会让我们了解技术将引领我们走向何方。
Our philosophy, both implicit and explicit, tries to avoid the two common traps: commitment to technological inevitability and commitment to strategies of incremental change. The technology itself will not draw us forward in any direction I can believe in either educationally or socially. The price of the education community’s reactive posture will be educational mediocrity and social rigidity. And experimenting with incremental changes will not even put us in a position to understand where the technology is leading.
我自己的哲学在变革概念上是革命性的，而不是改革性的。但我设想的革命是思想的革命，而不是技术的革命。它包括对特定学科领域的新理解和对学习过程本身的新理解。它包括对教育抱负的新而更雄心勃勃的视野。
My own philosophy is revolutionary rather than reformist in its concept of change. But the revolution I envision is of ideas, not of technology. It consists of new understandings of specific subject domains and in new understandings of the process of learning itself. It consists of a new and much more ambitious setting of the sights of educational aspiration.
我谈论的是思想革命，这种革命不能简单地归结为技术，就像物理学和分子生物学不能简单地归结为实验室使用的技术工具，诗歌不能简单地归结为印刷机一样。在我看来，技术有两个作用。一个是启发性的：计算机的存在催化了思想的出现。另一个是工具性的：计算机将把思想带入一个比迄今为止孵化思想的研究中心更大的世界。
I am talking about a revolution in ideas that is no more reducible to technologies than physics and molecular biology are reducible to the technological tools used in the laboratories or poetry to the printing press. In my vision, technology has two roles. One is heuristic: The computer presence has catalyzed the emergence of ideas. The other is instrumental: The computer will carry ideas into a world larger than the research centers where they have incubated up to now.
我曾指出，缺乏合适的技术是过去教育思想停滞不前的主要原因。大型计算机和现在的微型计算机的出现消除了这一停滞的原因。但还有另一个次要原因，它就像死水塘上的藻类一样不断滋生。我们必须考虑，它是否会随着允许其生长的条件而消失，或者像QWERTY一样，它是否会继续阻碍进步。为了定义这一障碍并正确看待它，我们将挑选出前面章节中提出的一个突出思想，并考虑除了技术之外，还需要什么来实现它。
I have suggested that the absence of a suitable technology has been a principle cause of the past stagnation of thinking about education. The emergence first of large computers and now of the microcomputer has removed this cause of stagnation. But there is another, secondary cause that grew like algae on a stagnant pond. We have to consider whether it will disappear with the condition that allowed its growth, or whether, like QWERTY, it will remain to strangle progress. In order to define this obstacle and place it in perspective, we shall pick out one of the salient ideas presented in earlier chapters and consider what besides technology is needed to implement it.
在计算概念和隐喻的熔炉中，在预测的广泛计算机能力和对儿童的实际实验中，皮亚杰学习的理念已成为一个重要的组织原则。从实践角度来看，这一理念制定了一个研究议程，旨在为儿童创造条件，让他们“自然地”探索以前需要说教的知识领域；也就是说，安排儿童接触他们可以用于皮亚杰学习的“材料”——物理的或抽象的。我们社会中成对事物的普遍存在就是“自然”发生的皮亚杰材料的一个例子。海龟环境为我们提供了“人造的”（即故意发明的）皮亚杰材料的例子。配对和海龟都将其数学能力归功于两个属性：儿童与它们相关，而它们又与重要的智力结构相关。因此，配对和海龟充当过渡对象。孩子们被吸引去玩配对游戏，并参与配对的过程，在这种游戏中，配对充当了强大思想的载体，或者说，强大思想将在孩子们活跃的思维矩阵中成长为萌芽。
Out of the crucible of computational concepts and metaphors, of predicted widespread computer power and of actual experiments with children, the idea of Piagetian learning has emerged as an important organizing principle. Translated into practical terms this idea sets a research agenda concerned with creating conditions for children to explore “naturally” domains of knowledge that have previously required didactic teaching; that is, arranging for the children to be in contact with the “material”—physical or abstract—they can use for Piagetian learning. The prevalence of paired things in our society is an example of “naturally” occurring Piagetian material. The Turtle environments gave us examples of “artificial” (that is, deliberately invented) Piagetian material. Pairings and Turtles both owe their mathetic power to two attributes: Children relate to them, and they in turn relate to important intellectual structures. Thus pairing and Turtles act as transitional objects. The child is drawn into playing with pairs and with the process of pairing and in this play pairing acts as a carrier of powerful ideas—or of the germs from which powerful ideas will grow in the matrix of the child’s active mind.
海龟与配对法共有的属性看似简单，但要实现这些属性，需要一套复杂的思想、各种专业知识和各种敏感性，虽然有些不自然，但可以分为三类：关于计算机的知识、关于学科领域的知识和关于人的知识。我认为设计好的皮亚杰材料所必需的人的知识本身就很复杂。它包括与学术心理学所有分支相关的知识——认知、人格、临床等等——以及创意艺术家和“与孩子相处融洽”的人所拥有的更具同理心的知识。在阐明创作皮亚杰材料的这些先决条件时，我们面对的是我所看到的关于计算机和教育未来的剩余基本问题：开发这些先决条件的人才供应问题。
The attributes the Turtle shares with pairing might seem simple, but their realization draws upon a complex set of ideas, of kinds of expertise, and of sensitivities that can be broken down, though somewhat artificially, into three categories: knowledge about computers, knowledge about subject domains, and knowledge about people. The people knowledge I see as necessary to the design of good Piagetian material is itself complex. It includes the kinds of knowledge that are associated with academic psychology in all its branches—cognitive, personality, clinical, and so on—and also the more empathetic kinds possessed by creative artists and by people who “get along with children.” In articulating these prerequisites for the creation of Piagetian material, we come face-to-face with what I see as the essential remaining problem in regard to the future of computers and education: the problem of the supply of people who will develop these prerequisites.
这个问题比单纯的人才短缺更为严重。过去，这类人没有发挥作用，这一事实已在社会和制度层面上得到体现；现在，他们有了作用，但没有容身之地。在当前的专业定义中，物理学家考虑如何做物理，教育家考虑如何教授物理。对于那些研究真正物理，但研究方向具有教育意义的人来说，他们没有公认的地位。这样的人在物理系并不特别受欢迎；他们的教育目标使他们的工作在其他物理学家眼中显得微不足道。他们在教育学院也不受欢迎——在那里，他们的高度技术性的语言不被理解，他们的研究标准也过时了。例如，在教育界，对“海龟微观世界”的新定理的评判标准是它是否对某一物理课程产生了“可衡量的”改进。我们假设的物理学家会以截然不同的方式看待他们的工作，认为这是对物理学的理论贡献，从长远来看，这将使物理宇宙的知识更容易获得，但从短期来看，这不会提高学生在物理课程中的表现。相反，如果将其作为局部变化注入基于不同理论方法的教育过程，甚至会损害学生的利益。
This problem goes deeper than a mere short supply of such people. The fact that in the past there was no role for such people has been cast into social and institutional concrete; now there is a role but there is no place for them. In current professional definitions physicists think about how to do physics, educators think about how to teach it. There is no recognized place for people whose research is really physics, but physics oriented in directions that will be educationally meaningful. Such people are not particularly welcome in a physics department; their education goals serve to trivalize their work in the eyes of other physicists. Nor are they welcome in the education school—there, their highly technical language is not understood and their research criteria are out of step. In the world of education a new theorem for a Turtle microworld, for example, would be judged by whether it produced a “measurable” improvement in a particular physics course. Our hypothetical physicists will see their work very differently, as a theoretical contribution to physics that in the long run will make knowledge of the physical universe more accessible, but which in the short run would not be expected to improve performance of students in a physics course. Perhaps, on the contrary, it would even harm the student if injected as a local change into an educational process based on a different theoretical approach.
关于教育学院和物理系欢迎什么样的讨论，这一点也更为普遍。资助机构和大学不会为任何过于深入地涉及科学思想的研究提供空间，因为这些研究不能归入教育的范畴，也不会过于深入地参与教育视角的研究，因为这些研究不能归入科学的范畴。似乎没有人会从根本上思考科学与人们思考和学习科学的方式之间的关系。尽管人们口头上强调了科学和社会的重要性，但其基本方法与传统教育的方法类似：将现成的科学要素传递给特定的受众。严肃地为人民创造科学的概念是相当陌生的。
This point about what kind of discourse is welcome in schools of education and in physics departments is true more generally also. Funding agencies as well as universities do not offer a place for any research too deeply involved with the ideas of science for it to fall under the heading of education and too deeply engaged in an educational perspective for it to fall under the heading of science. It seems to be nobody’s business to think in a fundamental way about science in relation to the way people think and learn it. Although lip service has been paid to the importance of science and society, the underlying methodology is like that of traditional education: one of delivering elements of ready-made science to a special audience. The concept of a serious enterprise of making science for the people is quite alien.
计算机本身无法改变现有的将科学家与教育家、技术专家与人文主义者区分开来的制度假设。它也无法改变关于科学是为人民服务的包装和交付问题还是严肃研究的适当领域的假设。要做到这些事情中的任何一件，都需要采取某种深思熟虑的行动，这种行动原则上可以在过去、在计算机出现之前发生。但这并没有发生。计算机提高了我们的不作为和行动的风险。对于那些希望看到改变的人来说，不作为的代价将是看到现状中最不理想的特征被夸大，甚至更加根深蒂固。另一方面，我们将处于一个快速发展的时期，这一事实将为制度变革提供立足点，而这在更稳定的时期可能是不可能实现的。
The computer by itself cannot change the existing institutional assumptions that separate scientist from educator, technologist from humanist. Nor can it change assumptions about whether science for the people is a matter of packaging and delivery or a proper area of serious research. To do any of these things will require deliberate action of a kind that could, in principle, have happened in the past, before computers existed. But it did not happen. The computer has raised the stakes both for our inaction and our action. For those who would like to see change, the price of inaction will be to see the least desirable features of the status quo exaggerated and even more firmly entrenched. On the other hand, the fact that we will be in a period of rapid evolution will produce footholds for institutional changes that might have been impossible in a more stable period.
电影作为一种新的艺术形式出现的同时，也催生出了一种新的亚文化，一种新的职业，由那些拥有前所未有的技能、敏感度和人生哲学的人组成。电影世界的发展史与人类社区的发展史密不可分。同样，一个新的个人电脑世界即将诞生，它的历史将与创造它的人的故事密不可分。
The emergence of motion pictures as a new art form went hand in hand with the emergence of a new subculture, a new set of professions made up of people whose skills, sensitivities, and philosophies of life were unlike anything that had existed before. The story of the evolution of the world of movies is inseparable from the story of the evolution of the communities of people. Similarly, a new world of personal computing is about to come into being, and its history will be inseparable from the story of the people who will make it.
结语
EPILOGUE
数学潜意识
The Mathematical Unconscious
重印于此作为结语的是我 几年前写的第一次讨论，该讨论发展成为本书的中心主题：我拒绝将刻板的“脱离实体”的数学与涉及人类各种敏感性的活动对立起来的二分法。我在书中以海龟几何为背景讨论了这个主题。在接下来的几页中，读者会发现这个主题嵌入在对数学乐趣来源的思考中。
REPRINTED HERE AS AN EPILOGUE IS MY FIRST DISCUSSION, WRITTEN a few years ago, of an idea that developed into a central theme of this book: My rejection of the dichotomy opposing a stereotypically “disembodied” mathematics to activities engaging a full range of human sensitivities.I In the book I discuss this theme in the context of Turtle geometry. In the following pages the reader will find this theme embedded in reflections on the sources of mathematical pleasure.
只有人类中的少数人，也许是极少数人，能够欣赏数学之美、体验数学之乐，这种观念在我们的文化中根深蒂固。亨利·庞加莱将这种观念视为理论原则，他不仅是本世纪最具开创性的数学思想家之一，也是数学科学认识论方面最有思想的作家之一，值得尊敬。庞加莱对数学家的看法与认知和教育心理学的流行趋势截然不同。对庞加莱来说，数学思维的显著特征不是逻辑性的，而是审美性的。他还认为，但这是另一个问题，这种美感是天生的：有些人天生就具有欣赏数学之美的能力，这些人可以成为富有创造力的数学家。其他人则不能。
It is deeply embedded in our culture that the appreciation of mathematical beauty and the experience of mathematical pleasure are accessible only to a minority, perhaps a very small minority, of the human race. This belief is given the status of a theoretical principle by Henri Poincaré, who has to be respected not only as one of the seminal mathematical thinkers of the century but also as one of the most thoughtful writers on the epistemology of the mathematical sciences. Poincaré differs sharply from prevalent trends in cognitive and educational psychology in his view of what makes a mathematician. For Poincaré the distinguishing feature of the mathematical mind is not logical but aesthetic. He also believes, but this is a separate issue, that this aesthetic sense is innate: Some people happen to be born with the faculty of developing an appreciation of mathematical beauty, and those are the ones who can become creative mathematicians. The others cannot.
本文以庞加莱的数学创造力理论为中心，对数学中逻辑与超逻辑的关系以及人类活动范围内数学与非数学的关系进行了反思。我们文化中的大众和高端两翼几乎一致地将这些二分法划分为清晰的界限。庞加莱的立场之所以格外有趣，是因为他在某些方面软化了这些界限，在某些方面又尖锐化了这些界限。当他将美学归因于数学中的重要功能作用时，这些界限就软化了。但是，假设一种特定的数学美学，特别是天生的美学，会加剧数学和非数学之间的分离。数学美学真的不同吗？它与我们美学体系的其他组成部分有共同的根源吗？数学的乐趣是源自其自身的快乐原则，还是源自那些激发人类生活其他阶段的快乐原则？数学直觉与常识的区别在于性质和形式，还是仅在内容上？
This essay uses Poincaré’s theory of mathematical creativity as an organizing center for reflections on the relationship between the logical and the extralogical in mathematics and on the relationship between the mathematical and the nonmathematical in the spectrum of human activities. The popular and the sophisticated wings of our culture almost unanimously draw these dichotomies in hard-edged lines. Poincare’s position is doubly interesting because in some ways he softens, and in some ways sharpens, these lines. They are softened when he attributes to the aesthetic an important functional role in mathematics. But the act of postulating a specifically mathematical aesthetic, and particularly an innate one, sharpens the separation between the mathematical and the nonmathematical. Is the mathematical aesthetic really different? Does it have common roots with other components of our aesthetic system? Does mathematical pleasure draw on its own pleasure principles, or does it derive from those that animate other phases of human life? Does mathematical intuition differ from common sense in nature and form or only in content?
这些问题深奥、复杂且古老。我之所以敢于在一篇短文中探讨这些问题，只是因为做了一些简化。首先是对问题的转化，其精神类似于让·皮亚杰将哲学问题转化为心理发生学问题，而对儿童思维方式的实验研究则变得令人耳目一新。通过这样做，他经常激怒或迷惑哲学家，但却极大地丰富了对心灵的科学研究。我的转化将庞加莱的最高数学创造力理论变成了一种更平凡但更易于理解的普通数学（也可能是非数学）思维理论。
These questions are deep, complex, and ancient. My daring to address them in the space of a short essay is justified only because of certain simplifications. The first of these is a transformation of the questions, similar in spirit to Jean Piaget’s way of transforming philosophical questions into psychogenetic ones to which experimental investigations into how children think become refreshingly relevant. By so doing, he has frequently enraged or bewildered philosophers, but has enriched beyond measure the scientific study of mind. My transformation turns Poincaré’s theory of the highest mathematical creativity into a more mundane but more manageable theory of ordinary mathematical (and possibly nonmathematical) thinking.
以这种方式将庞加莱的理论付诸实践可能会冒着抛弃庞加莱本人认为最重要的东西的风险。但这使该理论对心理学家、教育家和其他人来说更具有直接相关性，甚至可能非常紧迫。例如，如果庞加莱的模型包含了对普通数学思维的真实描述，那么今天的数学教育就完全是误入歧途，甚至是自欺欺人的。如果数学美学在学校里得到任何关注，那也只是作为一种附带现象，一种数学蛋糕上的糖霜，而不是使数学思维发挥作用的驱动力。当然，广泛实践的数学发展心理学理论（如皮亚杰的理论）完全忽视了美学，甚至直觉，而专注于数学思维逻辑方面的结构分析。
Bringing his theory down to earth in this way possibly runs the risk of abandoning what Poincaré himself might have considered to be most important. But it makes the theory more immediately relevant, perhaps even quite urgent, for psychologists, educators, and others. For example, if Poincaré’s model turned out to contain elements of a true account of ordinary mathematical thinking, it could follow that mathematical education as practiced today is totally misguided and even self-defeating. If mathematical aesthetics gets any attention in the schools, it is as an epiphenomenon, an icing on the mathematical cake, rather than as the driving force which makes mathematical thinking function. Certainly the widely practiced theories of the psychology of mathematical development (such as Piaget’s) totally ignore the aesthetic, or even the intuitive, and concentrate on structural analysis of the logical facet of mathematical thought.
当代数学教学的破坏性后果也可以看作是庞加莱的一个小悖论。学校和我们的文化总体上都远未培养儿童初生的数学审美意识，这一事实导致庞加莱关于美学重要性的主要论点破坏了他的次要论点的可信基础，该论点主张这种情感是天生的。如果庞加莱关于美学的观点是正确的，那么很容易看出，数学天赋的稀缺性如何能够不诉诸天赋来解释。
The destructive consequences of contemporary mathematics teaching can also be seen as a minor paradox for Poincaré. The fact that schools, and our culture generally, are so far from being nurturant of nascent mathematical aesthetic sense in children causes Poincaré’s major thesis about the importance of aesthetics to undermine the grounds for believing in his minor thesis, which asserts the innateness of such sensibilities. If Poincaré is right about aesthetics, it becomes only too easy to see how the apparent rareness of mathematical talent could be explained without appeal to innateness.
这些评论足以说明，即使庞加莱理论的平凡转变与大数学的工作过程完全脱节，对教育者来说也可能是一笔丰厚的奖励。但也许我们可以两全其美。通过采用一种更具体验性的讨论模式，数学思维理论可以立即与读者自己的心理过程进行对比，我们当然不会否认数学精英拥有类似经验的可能性。相反，庞加莱思想中在普通语境中最为明显有效的部分与现代趋势产生了强烈共鸣，在我看来，这些趋势构成了数学基础思维的范式转变。我的文章的最后几段将以布尔巴基数学结构理论为例，说明这种共鸣。
These remarks are enough to suggest that the mundane transformation of Poincaré’s theory might be a rich prize for educators even if it lost all touch with the processes at work in big mathematics. But perhaps we can have the best of both worlds. By adopting, as we shall, a more experiential mode of discussion through which theories about mathematical thinking can be immediately confronted with the reader’s own mental processes, we do not, of course, renounce the possibility that the mathematical elite share similar experiences. On the contrary, that part of Poincaré’s thinking which will emerge as most clearly valid in the ordinary context resonates strongly with modern trends which, in my view, constitute a paradigm shift in thinking about the foundations of mathematics. The concluding paragraphs of my essay will illustrate this resonance in the case of the Bourbaki theory of the structure of mathematics.
我的目的不是要提出一个具有清晰表述和严格论证的论点，当然也不是要对庞加莱理论的正确性作出判断。我将满足于（这是我的第二次重大简化）向非数学读者提出对数学的看法和关于数学的论述，这将使数学比通常的做法更接近他们所知道和享受的其他经验。这样做的主要障碍是数学的投影大大夸大了它的逻辑面貌，就像地球的墨卡托投影夸大了极地地区，以至于在地图上格陵兰岛北部比赤道巴西更壮观。因此，我们的讨论将旨在区分和关联我所说的数学的逻辑面貌和逻辑面貌。我将忽略应该在这些类别中做出的区分。数学之美、数学乐趣甚至数学直觉几乎可以互换对待，因为它们是逻辑面貌的代表。另一方面，我们不会将逻辑学中非常不同的方面分开，例如形式主义者强调演绎过程、伯特兰·罗素的还原论立场（庞加莱曾强烈反对这一立场）和阿尔弗雷德·塔斯基的集合论语义学。这些逻辑理论可以放在一起，因为它们都具有内在的、自主的数学观。它们将数学视为自足的，通过形式定义的（即数学的）有效性标准来证明自己，并且它们忽略了数学对自身之外的任何事物的引用。它们当然忽略了美和愉悦的现象。
My goal here is not to advance a thesis with crisp formulations and rigorous argument, and it is certainly not to pass judgment on the correctness of Poincaré’s theory. I shall be content (this is my second major simplification) to suggest to nonmathematical readers perceptions of, and a discourse about, mathematics which will place it closer than is commonly done to other experiences they know and enjoy. The major obstacle to doing so is a projection of mathematics which greatly exaggerates its logical face, much as the Mercator projection of the globe exaggerates the polar regions so that on the map northern Greenland becomes more imposing than equatorial Brazil. Thus our discussion will be aimed at distinguishing and relating what I shall call the extralogical face of mathematics and its logical face. I shall ignore distinctions which ought to be made within these categories. Mathematical beauty, mathematical pleasure, and even mathematical intuition will be treated almost interchangeably insofar as they are representatives of the extralogical. And, on the other side, we shall not separate such very different facets of the logical as the formalists’ emphasis on the deductive process, Bertrand Russell’s reductionist position (against which Poincaré fought so savagely), and Alfred Tarski’s set theoretic semantics. These logical theories can be thrown together insofar as they have in common an intrinsic, autonomous view of mathematics. They deal with mathematics as self-contained, as justifying itself by formally defined (that is, mathematical) criteria of validity, and they ignore all reference of mathematics to anything outside itself. They certainly ignore phenomena of beauty and pleasure.
只要不否认，数学逻辑学家就会忽略逻辑之外的东西，这一事实并不会引起理论上的矛盾。没有人会质疑数学逻辑的现实性，也不会质疑数学美或愉悦的现实性。庞加莱所挑战的，是理解数学活动、数学家的工作的可能性，仅仅，甚至主要是从逻辑的角度，而不参考美学。因此，他的挑战是在心理学或心理理论领域，因此，比理解数学思维这个看似专业的问题具有更广泛的反响：他的挑战质疑了心理学中认知功能的分离，这种分离是由它们与情感、感觉、美感的考虑相对立而定义的。
There is no theoretical tension in the fact that mathematical logicians ignore, as long as they do not deny, the extralogical. No one will call into question either the reality of the logical face of mathematics or the reality of mathematical beauty or pleasure. What Poincaré challenges is the possibility of understanding mathematical activity, the work of the mathematician solely, or even primarily, in logical terms without reference to the aesthetic. Thus his challenge is in the field of psychology, or the theory of mind, and, as such, has wider reverberations than the seemingly specialized problem of understanding mathematical thinking: His challenge calls into question the separation within psychology of cognitive functions, defined by their opposition to considerations of affect, of feeling, of sense of beauty.
总体而言，我站在庞加莱一边，反对数学思维“纯认知”理论的可能性，但对他赋予数学的高度特异性表示保留。但首先我必须介绍庞加莱理论的另一个主题。这就是潜意识的作用和性质。
I shall, on the whole, side with Poincaré against the possibility of a “purely cognitive” theory of mathematical thinking but express reservations about the high degree of specificity he attributes to the mathematical. But first I must introduce another of the themes of Poincaré’s theory. This is the role and the nature of the unconscious.
正如审美与逻辑的对立导致我们将庞加莱与认知心理学对立一样，潜意识与意识的对立导致庞加莱与弗洛伊德对立。庞加莱与弗洛伊德的观点相近，他明确地假设了两种心智（意识和潜意识），每种心智都受其自身的动态法则支配，每种心智都能够执行不同的功能，而对另一种心智活动的访问受到严重限制。我们将会看到，庞加莱对以下现象印象深刻：一个人早先一直在研究的问题的解决方案经常不经意间就进入意识，而且几乎是现成的，仿佛是由心智的某个隐藏部分产生的。但庞加莱的潜意识与弗洛伊德的潜意识截然不同。它远非前逻辑、充满性欲的原始过程的场所，而更像是一台情感中立、逻辑性极强的组合机器。
As the aesthetic versus the logical leads us to confront Poincaré with cognitive psychology, so the unconscious versus the conscious leads to a confrontation with Freud. Poincaré is close to Freud in clearly postulating two minds (the conscious and the unconscious) each governed by its own dynamic laws, each able to carry out different functions with severely limited access to the other’s activities. As we shall see, Poincaré is greatly impressed by the way in which the solution to a problem on which one has been working at an earlier time often comes into consciousness unannounced, and almost ready-made, as if produced by a hidden part of the mind. But Poincaré’s unconscious is very different from Freud’s. Far from being the site of prelogical, sexually charged, primary processes, it is rather like an emotionally neutral, supremely logical, combinatoric machine.
这些潜意识形象的对峙让我们回到了关于数学本身的本质的问题。数学的逻辑观在定义上是非有机的，脱离身体，只由纯粹和真理的内部逻辑塑造。这种观点与庞加莱的中性潜意识相一致，而不是与弗洛伊德的高度紧张、本能驱动的动力相一致。但正如我已经指出的那样，庞加莱本人拒绝了这种数学观；即使可以将其作为成品数学的形象（这已经很可疑），它作为数学真理和结构出现的生产过程的描述是完全不够的。在最简单的形式中，数学的逻辑形象是一个演绎系统，在这个系统中，新的真理是通过严格可靠的推理规则从以前得出的真理中得出的。虽然不那么天真的逻辑论者论点不能如此轻易地被推翻，但要注意这种数学解释可以被批评的不同方式是有意义的。它当然是不完整的，因为它未能解释选择过程，决定如何进行推论以及进行哪些推论。它具有误导性，因为数学家实际使用的推理规则如果应用不当，很快就会导致矛盾和悖论。最后，作为一种描述，它实际上是错误的，因为它没有为实际数学家花费最多时间的尚未调试的部分结果提供空间。数学工作不是沿着从真理到真理再到真理的狭窄逻辑路径进行的，而是勇敢地或摸索着在既不是完全正确也不是完全错误的命题的沼泽地中追寻偏差。
The confrontation of these images of the unconscious brings us back to our questions about the nature of mathematics itself. The logical view of mathematics is definitionally discorporate, detached from the body and molded only by an internal logic of purity and truth. Such a view would be concordant with Poincaré’s neutral unconscious rather than with Freud’s highly charged, instinct-ridden dynamics. But Poincaré himself, as I have already remarked, rejects this view of mathematics; even if it could be maintained (which is already dubious) as an image of the finished mathematical product, it is totally inadequate as an account of the productive process through which mathematical truths and structures emerge. In its most naive form, the logical image of mathematics is a deductive system in which new truths are derived from previously derived truths by means of rigorously reliable rules of inference. Although less naive logicist theses cannot be demolished quite so easily, it is relevant to notice the different ways in which this account of mathematics can be criticized. It is certainly incomplete since it fails to explain the process of choice determining how deductions are made and which of those made are pursued. It is misleading in that the rules of inference actually used by mathematicians would, if applied incautiously, quickly lead to contradictions and paradoxes. Finally, it is factually false as a description in that it provides no place for the as yet undebugged partial results with which the actual mathematician spends the most time. Mathematical work does not proceed along the narrow logical path of truth to truth to truth, but bravely or gropingly follows deviations through the surrounding marshland of propositions which are neither simply and wholly true nor simply and wholly false.
例如，人工智能领域的研究人员已经弥补了第一个薄弱环节，将设置和管理新问题的过程形式化为解决给定问题的一部分。但如果新问题及其生成规则以逻辑方式表达，我们认为这充其量是用动态逻辑取代静态逻辑。它并没有用其他东西取代逻辑。这里争论的问题是，即使在处理最纯粹的逻辑问题的过程中，数学家是否也会引发一些过程并设置一些本身并不纯粹逻辑的问题。
Workers in artificial intelligence have patched up the first of these areas of weakness, for example, by formalizing the process of setting and managing new problems as part of the work of solving a given one. But if the new problems and the rules for generating them are cast in logical terms, we see this as, at best, the replacement of a static logic by a dynamic one. It does not replace logic by something different. The question at issue here is whether even in the course of working on the most purely logical problem the mathematician evokes processes and sets problems which are not themselves purely logical.
偏离真理之路，陷入沼泽地的比喻虽然有些松散，但其优点在于，它尖锐地阐述了庞加莱的一个基本问题和关注点：引导问题，或者可以说是“在智力空间中导航”的问题。如果我们满足于大量产生逻辑后果，那么我们至少可以有一个安全的过程。事实上，根据庞加莱的说法，数学家受到审美意识的引导：在工作中，数学家经常必须处理不同程度错误的命题，但不必考虑任何冒犯个人数学美感的命题。
The metaphor of wandering off the path of truth into surrounding marshlands has the merit, despite its looseness, of sharply stating a fundamental problem and preoccupation of Poincaré’s: the problem of guidance, or one might say, of “navigation in intellectual space.” If we are content to churn out logical consequences, we would at least have the security of a safe process. In reality, according to Poincaré, the mathematician is guided by an aesthetic sense: In doing a job, the mathematician frequently has to work with propositions which are false to various degrees but does not have to consider any that offend a personal sense of mathematical beauty.
庞加莱关于审美如何指导数学工作的理论将工作分为三个阶段。第一阶段是深思熟虑的、有意识的分析阶段。根据庞加莱的说法，如果问题很难，第一阶段永远不会产生解决方案。它的作用是创建解决方案的元素。必须有一个无意识的工作阶段介入，在数学家看来，这似乎是暂时放弃任务或让问题孵化。庞加莱假设了一种孵化机制。现象学的放弃观点是完全错误的。相反，问题被交给了一个非常活跃的潜意识，它坚持不懈地开始结合工作的第一个有意识状态提供给它的元素。除了集中注意力、系统运作和不受无聊、干扰或目标改变的影响外，潜意识被认为没有任何非凡的力量。潜意识工作的成果在与后者正在做的事情无关的时刻被送回意识。这一次，现象学的观点更加具有误导性，因为完成的作品可能会在最令人惊讶的时刻出现在意识中，显然与相当偶然的事件有关。
Poincaré’s theory of how the aesthetic guides mathematical work divides the work into three stages. The first is a stage of deliberate, conscious analysis. If the problem is difficult, the first stage will never, according to Poincaré, yield the solution. Its role is to create the elements out of which the solution will be constructed. A stage of unconscious work, which might appear to the mathmatician as temporarily abandoning the task or leaving the problem to incubate, has to intervene. Poincaré postulates a mechanism for the incubation. The phenomenological view of abandonment is totally false. On the contrary, the problem has been turned over to a very active unconscious which relentlessly begins to combine the elements supplied to it by the first, conscious state of the work. The unconscious mind is not assumed to have any remarkable powers except concentration, systematic operation, and imperviousness to boredom, distractions, or changes of goal. The product of the unconscious work is delivered back to the conscious mind at a moment which has no relation to what the latter is doing. This time the phenomenological view is even more misleading since the finished piece of work might appear in consciousness at the most surprising times, in apparent relation to quite fortuitous events.
潜意识如何知道该把什么传回给意识？庞加莱正是在这里看到了审美的作用。他认为，从经验观察来看，传回的想法不一定是原始问题的正确解决方案。因此，他得出结论，潜意识无法严格确定一个想法是否正确。但被传回的想法总是带有数学美的印记。这项工作第三阶段的功能是有意识地、严格地检查从潜意识中获得的结果。它们可能会被接受、修改或拒绝。在最后一种情况下，潜意识可能会再次发挥作用。我们观察到，该模型假设除了意识和潜意识之外还有第三个代理。这个代理有点类似于弗洛伊德的审查员；它的工作是扫描不断变化的万花筒般的潜意识模式，只允许那些满足其审美标准的模式通过心灵之间的门户。
How does the unconscious mind know what to pass back to the conscious mind? It is here where Poincaré sees the role of the aesthetic. He believes, as a matter of empirical observation, that ideas passed back are not necessarily correct solutions to the original problem. So he concludes that the unconscious is not able to rigorously determine whether an idea is correct. But the ideas passed up do always have the stamp of mathematical beauty. The function of the third stage of the work is to consciously and rigorously examine the results obtained from the unconscious. They might be accepted, modified, or rejected. In the last case the unconscious might once more be called into action. We observe that the model postulates a third agent in addition to the conscious and unconscious minds. This agent is somewhat akin to a Freudian censor; its job is to scan the changing kaleidoscope of unconscious patterns, allowing only those which satisfy its aesthetic criteria to pass through the portal between the minds.
庞加莱描述的是数学创造力的最高水平，我们不能假设更基本的数学工作也遵循相同的动态过程。但在我们自己努力建立数学思维理论的过程中，我们也不应该假设相反的情况，因此，看到庞加莱所描述的过程与我们要求非数学家在麻省理工学院 ( MIT)进行的数学问题研究中所展示的模式之间哪怕是极其有限的结构相似性，也是令人鼓舞的。这项研究被称为“大声思考”，是一组旨在引发富有成效的思考（通常在他们通常会避免的领域，例如数学）并尽可能明确地表达这些思考的技术。以下示例说明了最简单的审美思维指导可能是什么。实验中的受试者显然按照庞加莱在第二阶段所假设的组合方式进行，直到获得令人满意的结果，其理由至少在审美和逻辑上都令人满意。这个过程与庞加莱的描述不同，因为它仍然处于意识层面。这可以从很多方面与庞加莱的理论相协调：有人可能会认为，产生可接受结果所需的组合动作数量太少，不需要将问题传递到无意识层面，或者这些非数学家缺乏无意识地完成此类工作的能力。无论如何，这个例子（实际上是整篇文章）的重点不是详细捍卫庞加莱，而是说明审美指导的概念。
Poincaré is describing the highest level of mathematical creativity, and one cannot assume that more elementary mathematical work follows the same dynamic processes. But in our own striving toward a theory of mathematical thinking we should not assume the contrary either, and so it is encouraging to see even very limited structural resemblances between the process as described by Poincaré and patterns displayed by nonmathematicians whom we asked to work on mathematical problems in what has come, at MIT, to be called “Loud Thinking,” a collection of techniques designed to elicit productive thought (often in domains, such as mathematics, they would normally avoid) and make as much of it as possible explicit. The example that follows illustrates aspects of what the very simplest kind of aesthetic guidance of thought might be. The subjects in the experiment clearly proceed by a combinatoric, such as that which Poincaré postulates in his second stage, until a result is obtained which is satisfactory on grounds that have at least as much claim to be called aesthetic as logical. The process does differ from Poincaré’s description in that it remains on the conscious level. This could be reconciled with Poincaré’s theory in many ways: One might argue that the number of combinatorial actions needed to generate the acceptable result is too small to require passing the problem to the unconscious level, or that these nonmathematicians lack the ability to do such work unconsciously. In any case, the point of the example (indeed, of this essay as a whole) is not to defend Poincaré in detail but to illustrate the concept of aesthetic guidance.
要求受试者解决的问题是证明 2 的平方根是无理数。这里的选择特别合适，因为英国数学家 GH Hardy 选择这个定理作为数学之美的典范，因此，在非精英主义的数学美学讨论中，有趣的是，如果情感支持的工作环境鼓励他们继续前进，尽管数学知识有限，但许多数学知识很少的人仍然能够找到证明。以下段落描述了我们调查中的几乎所有受试者都经历过的一个情节。为了将我们自己投射到这个情节中，让我们假设我们已经建立了等式：
The problem on which the subjects were asked to work was the proof that the square root of 2 is irrational. The choice is particularly appropriate here because this theorem was selected by the English mathematician G. H. Hardy as a prime example of mathematical beauty, and consequently it is interesting, in the context of a nonelitist discussion of mathematical aesthetics, to discover that many people with very little mathematical knowledge are able to discover the proof if emotionally supportive working conditions encourage them to keep going despite mathematical reticence. The following paragraphs describe an episode through which almost all the subjects in our investigation passed. To project ourselves into this episode, let us suppose that we have set up the equation:
其中p和q是整数
where p and q are whole numbers
让我们假设我们真的不相信可以这样表达。为了证明这一点，我们试图揭示方程式难以理解的纯真表面印象背后一些奇怪、实际上矛盾的东西。我们显然与潜在内容和显性内容的相互作用有关。在这种情况下，哪些步骤有帮助？
Let us also suppose that we do not really believe that can be so expressed. To prove this, we seek to reveal something bizarre, in fact contradictory, behind the impenetrably innocent surface impression of the equation. We clearly have to do with an interplay of latent and manifest contents. What steps help in such cases?
就像读过弗洛伊德的书一样，许多受试者都参与了数学“自由联想”的过程，依次尝试与此类方程相关的各种变换。数学更精湛的人需要的尝试次数较少，但似乎没有一个受试者会受到工作将走向何方的预测的指导。以下是一些变换示例，按受试者进行变换的顺序排列：
Almost as if they had read Freud, many subjects engage in a process of mathematical “free association,” trying in turn various transformations associated with equations of this sort. Those who are more sophisticated mathematically need a smaller number of tries, but none of the subjects seem to be guided by a prevision of where the work will go. Here are some examples of transformations in the order they were produced by one subject:
p2 = 2q2
p2 = 2q2
所有对这个问题进行过深入研究的受试者在想到最后一个等式时都会表现出明显的兴奋和愉悦。这种愉悦并不依赖于知道（至少是有意识地知道）这个过程将导致什么结果。它发生在受试者能够说出下一步要做什么之前，事实上，即使在根本没有取得进一步进展的情况下，它也会这样做。对p 2 = 2 q 2的反应不仅仅是情感上的；一旦看到这一点，受试者几乎就不会回头看之前的任何变换，甚至不会再看原始方程式。因此，p 2 = 2 q 2有一些非常特别之处。是什么呢？我们首先集中讨论它无疑具有愉悦感这一事实，并推测这种愉悦的来源。愉悦在数学中扮演什么角色？
All subjects who have become more than very superficially involved in the problem show unmistakable signs of excitement and pleasure when they hit on the last equation. This pleasure is not dependent on knowing (at least consciously) where the process is leading. It happens before the subjects are able to say what they will do next, and, in fact, it happens even in cases where no further progress is made at all. And the reaction to p2 = 2q2 is not merely affective; once this has been seen, the subjects scarcely ever look back at any of the earlier transformations or even at the original equations. Thus there is something very special about p2 = 2q2. What is it? We first concentrate on the fact that it undoubtedly has a pleasurable charge and speculate about the sources of the charge. What is the role of pleasure in mathematics?
当然，在数学工作中，人们常常会体验到快乐，就好像一个人在经过艰苦奋斗后实现了所期望的目标时在奖励自己。但是，这个实际的等式在这里被预期为一个预先设定的目标，这是非常不可信的。如果快乐是实现目标的快乐，那么这个目标就与实现一个特定的等式有着非常不同、不那么正式的性质，我会说“更具审美性”。要确切地知道它是什么，需要对各个受试者有比我们在这里所能包括的更多的了解。它当然因受试者而异，甚至在每个受试者身上都存在多重决定。一些受试者明确为自己设定了目标：“去掉平方根。”其他受试者似乎没有明确为自己设定这个目标，但仍然很高兴看到平方根符号消失。同样，其他人对 2 = p 2 / q 2的出现没有做出任何特殊反应，直到它变成p 2 = 2 q 2。我认为，为了显而易见的、简单的工具性目的而消除根符号只是一个更复杂故事的一部分：事件与几个过程产生共鸣，这些过程可能或不可能被意识所理解，也可能或不可能被明确地表述为目标。我还认为，其中一些过程会利用其他快乐来源，这些快乐来源比一般的目标实现更具体，甚至可能更原始。为了使这些建议更加具体，我将给出两个此类快乐过程的例子。
Pleasure is, of course, often experienced in mathematical work, as if one were rewarding oneself when one achieves a desired goal after arduous struggle. But it is highly implausible that this actual equation was anticipated here as a preset goal. If the pleasure was that of goal achievement, the goal was of a very different, less formal, I would say “more aesthetic” nature than the achievement of a particular equation. To know exactly what it is would require much more knowledge about the individual subjects than we can include here. It is certainly different from subject to subject and even multiply overdetermined in each subject. Some subjects explicitly set themselves the goal: “Get rid of the square root.” Other subjects did not seem explicitly to set themselves this goal but were nevertheless pleased to see the square root sign go away. Others, again, made no special reaction to the appearance of 2 = p2/q2 until this turned into p2 = 2q2. My suggestion is that the elimination of the root sign for the obvious, simple, instrumental purpose is only part of a more complex story: The event is resonant with several processes which might or might not be accessible to the conscious mind and might or might not be explicitly formulated as goals. I suggest, too, that some of these processes tap into other sources of pleasure, more specific and perhaps even more primitive than the generalized one of goal attainment. To make these suggestions more concrete, I shall give two examples of such pleasure-giving processes.
第一个例子最好用情境演算的格框架类型来描述，这种演算表征了人工智能领域的最新思想。原始方程式被形式化为一个情境框架，其中有“三个参与者”的格槽，其中主要或“主题”参与者是参与者。另外两个参与者p和q是从属的虚拟参与者，其角色仅仅是对主题参与者做出断言。当我们将情境转变为p 2 = 2 q 2 时，它与婴儿对躲猫猫的感知中的图形/背景反转或用脸代替屏幕一样截然不同。现在p已成为主题，而先前的主题已消失。这是否会利用使婴儿普遍喜欢躲猫猫的快乐源泉？
The first example is best described in terms of the case frame type of calculus of situations characterizing recent thinking in artificial intelligence. The original equation is formalized as a situation frame with case slots for “three actors,” of which the principle or “subject” actor is. The other two actors, p and q, are subordinate dummy actors whose roles are merely to make assertions about the subject actor. When we turn the situation into p2 = 2q2, it is as sharply different as in a figure/ground reversal or the replacement of a screen by a face in an infant’s perception of peek-a-boo. Now p has become the subject, and the previous subject has vanished. Does this draw on the pleasure sources that make infants universally enjoy peek-a-boo?
这一过程中另一个令人愉悦的例子是 2 并没有完全消失得无影无踪。在p 2 = 2 q 2中 2 仍然可见！然而，这两次出现的 2 的作用截然不同，识别它们会给情境带来双关语或“凝缩”的性质，至少有点像弗洛伊德所认为的对诙谐的有效性至关重要的凝缩。这一建议的吸引力和合理性来自于在很多数学情境中都能看到凝缩的可能性。事实上，抽象数学的核心思想就可以看作是凝缩：“抽象”描述同时表示非常不同的“具体”事物。这是否可以让我们推测，数学与笑话、梦和歇斯底里有比人们通常认识到的更多的共同之处？
The other example of what might be pleasing in this process comes from the observation that 2 has not vanished completely without trace. The 2 is still visible in p2 = 2q2! However, these two occurences of 2 are so very different in role that identifying them gives the situation a quality of punning, or “condensation” at least somewhat like that which Freud sees as fundamental to the effectiveness of wit. The attractiveness and plausibility of this suggestion comes from the possibility of seeing condensation in very many mathematical situations. Indeed, the very central idea of abstract mathematics could be seen as condensation: The “abstract” description simultaneously signifies very different “concrete” things. Does this allow us to conjecture that mathematics shares more with jokes, dreams, and hysteria than is commonly recognized?
当然，过于强调p 2 = 2 q 2的优点而脱离其在实现最初目的中的作用是危险的，最初的目的不是为了激起数学快感，而是为了证明 2 是无理数。前两段的陈述需要与对作品如何通过一个与将其视为证明定理的超目标的子目标并非完全独立的过程来关注p 2 = 2 q 2的理解相结合。我们如何将功能与美学结合起来？对于那些将功能性极强的子目标系统视为主要推动力的人来说，最简单的做法是扩大可以制定子目标的论域。在适当的情境框架系统中，将问题场景中的从属角色（即p）提升为主要角色，就像找到方程的数值解一样，是一个定义明确的子目标。但我们现在谈论的目标已经失去了数学特性，可能与生活或文学中的非数学情境共享。这种思路如果发展到极端，就会让我们把数学，甚至其细节，看作是其他事物的演绎：演员可能是数学对象，但情节是用其他术语来表达的。即使在不那么极端的形式中，这也表明了审美和功能可以形成一种共生关系，可以说是相互利用。数学上的功能目标是通过在另一个非数学话语中制定的子目标的发挥来实现的，这些子目标借鉴了相应的数学以外的知识。因此，功能利用了审美。但在某种程度上，我们（在这里以一种非常弗洛伊德的精神）认为数学过程本身就是前数学过程的演绎，反之亦然。
It is of course dangerous to go too far in the direction of presenting the merits of p2 = 2q2 in isolation from its role in achieving the original purpose, which was not to titillate the mathematical pleasure senses but to prove that 2 is irrational. The statement of the previous two paragraphs needs to be melded with an understanding of how the work comes to focus on p2 = 2q2 through a process not totally independent of recognizing it as a subgoal of the supergoal of proving the theorem. How do we integrate the functional with the aesthetic? The simplest gesture in this direction for those who see the eminently functional subgoal system as the prime mover is to enlarge the universe of discourse in which subgoals can be formulated. Promoting a subordinate character (that is, p) on the problem scene to a principal role is, within an appropriate system of situation frames, as well-defined a subgoal as, say, finding the numerical solution of an equation. But we are now talking about goals which have lost their mathematical specificity and may be shared with nonmathematical situations of life or literature. Taken to its extreme, this line of thinking leads us to see mathematics, even in its detail, as an acting out of something else: The actors may be mathematical objects, but the plot is spelled out in other terms. Even in its less extreme forms this shows how the aesthetic and the functional can enter into a symbiotic relationship of, so to speak, mutual exploitation. The mathematically functional goal is achieved through a play of subgoals formulated in another, nonmathematical discourse, drawing on corresponding extramathematical knowledge. Thus the functional exploits the aesthetic. But to the extent we see (here in a very Freudian spirit) the mathematical process itself as acting out premathematical processes, the reverse is also true.
这些推测在一定程度上（非常有限地）表明了庞加莱的数学美学警戒如何能够与现有的思维模式相协调，从而丰富两者。但是，这样做的尝试非常尖锐地提出了一个基本问题，即不仅数学，而且所有智力工作的功能性与美学和享乐主义方面之间的关系。是什么让它们能够为彼此服务？知识或欣赏原则在数学之外有用，却可以在数学内部应用，这难道不奇怪吗？答案一定在于数学的发生理论。如果我们采用柏拉图式（或逻辑）的观点，认为数学独立于人类思维或人类活动的任何属性而存在，我们就不得不认为这种解释极不可能。在剩下的篇幅中，我将再举几个例子，说明如何从一种使其与其他人类结构的关系更加自然的角度看待数学。我们首先来看关于 2 的平方根的故事的另一个情节。
These speculations go some (very little) way toward showing how Poincaré’s mathematical aesthetic sentinel could be reconciled with existing models of thinking to the enrichment of both. But the attempt to do so very sharply poses one fundamental question about the relationship between the functional and the aesthetic and hedonistic facets not only of mathematics but of all intellectual work. What is it about each of these that makes it able to serve the other? Is it not very strange that knowledge, or principles of appreciation, useful outside of mathematics, should have application within? The answer must lie in a genetic theory of mathematics. If we adopt a Platonic (or logical) view of mathematics as existing independently of any properties of the human mind, or of human activity, we are forced to see such interpretations as highly unlikely. In the remaining pages I shall touch on a few more examples of how mathematics can be seen from a perspective which makes its relationship to other human structures more natural. We begin by looking at another episode of the story about the square root of 2.
我们对p 2 = 2 q 2的讨论几乎是完全非目的论的，因为我们只从一个方面，即它起源的方面，来讨论它，假装不知道它要去哪里。现在我们通过看它如何服务于这项工作的初衷来弥补这一点，即在假设中寻找矛盾。人们可以采取多种途径来实现这一目标。我将对比其中的两种，它们在一个维度上有所不同，人们可以称之为“格式塔与原子论”或“一闪而过的洞察力与逐步推理”。逐步形式是更经典的形式（它归功于欧几里得本人），并以以下方式进行。我们可以从p 2 = 2 q 2中得出p 2是偶数。由此可知p是偶数。根据定义，这意味着p是另一个整数的两倍，我们可以将其称为r。所以：
Our discussion of p2 = 2q2 was almost brutally nonteleological in that we discussed it from only one side, the side from which it came, pretending ignorance of where it was going. We now remedy this by seeing how it serves the original intention of the work, which was to find a contradiction in the assumption. It happens that there are several paths one can take to this goal. Of these I shall contrast two which differ along a dimension one might call “gestalt versus atomistic” or “aha-single-flash-insight versus step-by-step reasoning.” The step-by-step form is the more classical (it is attributed to Euclid himself) and proceeds in the following manner. We can read off from p2 = 2q2 that p2 is even. It follows that p is even. By definition this means that p is twice some other whole number which we can call r. So:
p = 2 r
p = 2r
p2 = 4r2
p2 = 4r2
2 q 2 = 4 r 2记住：p 2 = 2 q 2（！）
2q2 = 4r2 remember: p2 = 2q2(!)
q2 = 2r2
q2 = 2r2
我们推断q也是偶数。但这最终确实显得很奇怪，因为我们一开始就选择了p和q，如果我们愿意的话，可以确保它们没有共同因子。所以存在矛盾。
and we deduce that q is also even. But this at last really is manifestly bizarre since we chose p and q in the first place and could, had we wished, have made sure that they had no common factor. So there is a contradiction.
在评论这一过程的美学性之前，我们先来看看“闪现”版的证明。它依赖于对整数的某种感知，即作为质因数的唯一集合：6 = 3 × 2 和 36 = 3 × 3 × 2 × 2。如果你拥有这种感知数字的框架，你可能会立即感觉到一个完全平方数（36 或p 2或q 2）是一个偶数集。如果你不具备这种框架，我们可能不得不使用分步论证（例如让p = p 1 p 2 … p k，从而p 2 = p 1 p 1 p 2 p 2 … p k p k），然后这个证明就会变得更加原子化，当然也不会像经典形式那样令人愉悦。但是，如果你确实将p 2和q 2视为（或训练自己将其视为）偶数集，你也会认为p 2 = 2 q 2做出了一个荒谬的断言，即偶数集（p 2）等于奇数集（q 2加上一个附加因子：2）。因此，给定正确的数字感知框架，p 2 = 2 q 2被直接视为荒谬的（或者从现象学上看是这样）。
Before commenting on the aesthetics of this process, we look at the “flash” version of the proof. It depends on having a certain perception of whole numbers, namely, as unique collections of prime factors: 6 = 3 × 2 and 36 = 3 × 3 × 2 × 2. If you solidly possess this frame for perceiving numbers, you probably have a sense of immediate perception of a perfect square (36 or p2 or q2) as an even set. If you do not possess it, we might have to use step-by-step arguments (such as let p = p1 p2… pk, so that p2 = p1 p1 p2 p2… pk pk), and this proof then becomes even more atomistic and certainly less pleasing than the classical form. But if you do see (or train yourself to see) p2 and q2 as even sets, you will also see p2 = 2q2 as making the absurd assertion that an even set (p2) is equal to an odd set (q2 and one additional factor: 2). Thus given the right frames for perceiving numbers, p2 = 2q2 is (or so it appears phenomenologically) directly perceived as absurd.
虽然关于这两个小证明的美学比较有很多话要说，但我将集中讨论一些受试者在我们的实验中发现的美和愉悦的一个方面。许多人对第二个证明的出色表现印象深刻。但如果后者以其聪明和直接性而引人注目，那并不意味着第一个证明因为（在我看来）本质上是连续的而失败。相反，在连续过程中，人们被捕获和不可阻挡地进行的方式非常有力。我的意思不仅仅是当另一个人很好地呈现证明时，证明在修辞上具有说服力，尽管这是数学观赏性方面的一个重要因素。我的意思是，你只需要很少的数学知识就可以强制移动这些步骤，这样一旦你开始走上正轨，你就会发现你生成了整个证明。
Although there is much to say about the comparative aesthetics of these two little proofs, I shall concentrate on just one facet of beauty and pleasure found by some subjects in our experiments. Many people are impressed by the brilliance of the second proof. But if this latter attracts by its cleverness and immediacy, it does not at all follow that the first loses by being (as I see it) essentially serial. On the contrary, there is something very powerful in the way one is captured and carried inexorably through the serial process. I do not merely mean that the proof is rhetorically compelling when presented well by another person, although this is an important factor in the spectator sport aspect of mathematics. I mean rather that you need very little mathematical knowledge for the steps to be forced moves, so that once you start on the track you will find that you generate the whole proof.
人们可以用非常不同的方式和非常不同的情感来体验必然性的过程。人们可以体验到它被接管为一种暂时的服从关系。人们可以体验到它对数学的屈服，或对另一个人的屈服，或将自己的一部分屈服于另一部分。人们可以体验到它不是屈服，而是在行使一种令人振奋的力量。这些都可以体验到美丽、丑陋、愉悦、厌恶或恐惧。
One can experience the process of inevitability in very different ways with very different kinds of affect. One can experience it as being taken over in a relationship of temporary submission. One can experience this as surrender to mathematics, or to another person, or of one part of oneself to another. One can experience it not as submission but as the exercise of an exhilarating power. Any of these can be experienced as beautiful, as ugly, as pleasurable, as repulsive, or as frightening.
这些评论虽然只是现象的表面，但足以让人严重怀疑庞加莱相信数学审美能力是天生的，独立于其他思维成分的理由。这些评论表明，庞加莱没有考虑到的因素在很多方面可能在原则上对一个人认为数学是美的还是丑的以及他特别喜欢或讨厌哪种数学产生强大的影响。为了更清楚地了解这些因素，让我们暂时离开数学，来看一部非常敏感的小说中的一个例子：罗伯特·波西格的《禅与摩托车维修艺术》。这本书是一部关于不同思维风格的哲学小说。叙述事件的主角和他的朋友约翰·萨瑟兰正在骑摩托车度假，从东海岸骑到蒙大拿州。在书中叙述的这次旅行之前不久，约翰·萨瑟兰曾提到他的车把打滑了。叙述者很快决定需要做一些垫片，并建议从铝制啤酒罐上切下垫片。“我自己也认为这个主意很聪明，”他说道，并描述了他对萨瑟兰的反应感到震惊，这种反应几乎导致他们的友谊破裂。对萨瑟兰来说，这个想法一点也不聪明；它是无法形容的冒犯。叙述者解释说：“我竟然厚颜无耻地建议用一块旧啤酒罐来修理他那辆新买的价值 1800 美元的宝马，这辆车是半个世纪以来德国机械工艺的骄傲！”但对于叙述者来说，这并不矛盾；相反：“啤酒罐铝是软而粘的金属。非常适合这种应用...换句话说，任何一个拥有半个世纪机械工艺经验的真正的德国机械师都会认为，这个针对特定技术问题的特定解决方案是完美的。”事实证明，这种差异是无法弥合的，并且会让人情绪崩溃。尽管这两个朋友彼此之间以及与他们的摩托车之间关系非常亲密，可以一起踏上书中描述的漫长旅程，但他们之间达成的默契使他们不再讨论摩托车的保养和维修，从而挽救了这段友谊。
These remarks, although they remain at the surface of the phenomenon, suffice to cast serious doubt on Poincaré’s reasons for believing that the faculty for mathematical aesthetic is inborn and independent of other components of the mind. They suggest too many ways in which factors of a kind Poincaré does not consider might, in principle, powerfully influence whether an individual finds mathematics beautiful or ugly and which kinds of mathematics he will particularly relish or revile. To see these factors a little more clearly, let us leave mathematics briefly to look at an example from a very sensitive work of fiction: Robert Pirsig’s Zen and the Art of Motorcycle Maintenance. The book is a philosophical novel about different styles of thought. The principal character, who narrates the events, and his friend John Sutherland are on a motorcycling vacation which begins by riding from the east coast to Montana. Some time before the trip recounted in the book, John Sutherland had mentioned that his handlebars were slipping. The narrator soon decided that some shimming was necessary and proposed cutting shim stock from an aluminum beer can. “I thought this was pretty clever myself,” he says, describing his surprise at Sutherland’s reaction which brought the friendship close to rupture. To Sutherland the idea was far from clever; it was unspeakably offensive. The narrator explains: “I had had the nerve to propose repair of his new eighteen-hundred-dollar BMW, the pride of a half-century of German mechanical finesse, with a piece of old beer can!” But for the narrator there is no conflict; on the contrary: “Beer can aluminum is soft and sticky as metals go. Perfect for the application… in other words any true German mechanic with half a century of mechanical finesse behind him, would have concluded that this particular solution to this particular technical problem was perfect.” The difference proves to be unbridgeable and emotionally explosive. The friendship is saved only by a tacit agreement never again to discuss maintenance and repair of the motorcycles even though the two friends are close enough to one another and to their motorcycles to embark together on the long trip described in the book.
如果萨瑟兰的反应表现出愚蠢、无知或对临时解决问题的特殊癖好，那么对于我们的问题来说，这种反应是无关紧要的。但它的意义比这些都深刻。波西格的成就是向我们展示了许多此类事件中的连贯性。这一成就令人印象深刻。波西格向我们提供了丰富的材料，我们可以用它们来欣赏事件中隐含的各种连贯性，这些连贯性与波西格自己提出的连贯性大不相同。在这里，我想简要谈谈萨瑟兰和垫片的故事与我们讨论过的数学问题之间的两个类比：首先，在思考数学和摩托车时美学与逻辑之间的关系，其次，数学或摩托车与其他一切之间的连续性和不连续性。
Sutherland’s reaction would be without consequence for our problem if it showed stupidity, ignorance, or an idiosyncratic quirk about ad hoc solutions to repair problems. But it goes deeper than any of these. Pirsig’s accomplishment is to show us the coherence in many such incidents. This accomplishment is quite impressive. Pirsig presents us with materials so rich that we can use them to appreciate kinds of coherence implicit in the incidents which are rather different from the one advanced by Pirsig himself. Here I want to touch briefly on two analogies between the story of Sutherland and the shim stock and issues we have discussed about mathematics: first, the relationship between aesthetics and logic in thinking about mathematics as well as motorcycles, and second, the lines of continuity and discontinuity between mathematics or motorcycles and everything else.
从垫片事件本身，以及本书的其余部分可以清楚地看出，对于波西格的每个角色来说，人、机器和自然环境之间的连续性都非常不同，而且这些差异深深影响了他们的审美。对于叙述者来说，摩托车不仅与啤酒罐的世界相连，而且更普遍地与金属（作为物质）的世界相连。在这个世界上，金属的身份不能简化为摩托车或啤酒罐中金属的特定体现。任何身份也不能简化为它的特定实例。相反，对于萨瑟兰来说，这种连续性不仅是无形的，而且他非常重视维持叙述者所看到的同一种物质的表面表现之间的界限。
It is clear from the shim stock incident itself, and much more so from the rest of the book, that the continuity between man, machine, and natural environment is very different for each of Pirsig’s characters and that these differences deeply affect their aesthetic appreciation. For the narrator, the motorcycle is continuous with the world not only of beer cans but more generally the world of metals (taken as substance). In this world, the metal’s identity is not reducible to a particular embodiment of the metal in a motorcycle or in a beer can. Nor can any identity be reduced to a particular instance of it. For Sutherland, on the contrary, this continuity is not merely invisible, but he has a strong investment in maintaining the boundaries between what the narrator sees as superficial manifestations of the same substance.
对于萨瑟兰来说，摩托车不仅与啤酒罐截然不同，甚至与其他机器也截然不同，这一事实使他能够毫无冲突地将这项技术视为逃避技术的手段。我们可以通过注意这两个人物在工作和社会中的不同参与来深入分析他们在各自位置上的投入。叙述者是工业社会的一部分（他在一家计算机公司工作），被迫寻找自己的身份（就像他寻找金属的身份一样），他的本质超越了他被塑造成的特定形式。就像可锻造的金属一样，他超越了现在强加给他的形式，甚至可能比现在强加的形式更好。他当然不会将自己定义为计算机手册的作者。另一方面，他的朋友萨瑟兰是一名音乐家，他更能将他的作品视为构成他自我形象的东西，就像他把摩托车当作摩托车，把啤酒罐当作啤酒罐一样。
For Sutherland, the motorcycle is not only in a world apart from beer cans; it is even in a world apart from other machines, a fact that enables him to relate without conflict to this piece of technology as a means to escape from technology. We could deepen the analysis of the investments of these two characters in their respective positions by noting their very different involvements in work and society. The narrator is part of industrial society (he works for a computer company) and is forced to seek his own identity (as he seeks the identity of metal) in a sense of his substance which lies beyond the particular form into which he has been molded. Like malleable metal, he is something beyond and perhaps better than the form which is now imposed on him. He certainly does not define himself as a writer of computer manuals. His friend Sutherland, on the other hand, is a musician and is much more able to take his work as that which structures his image of himself in the same way that he takes a motorcycle as a motorcycle and a beer can as a beer can.
我们不需要进一步探究这些本质和偶然的问题就能阐明一个重要的观点，而这个观点却被人们广泛忽视：如果摩托车维修的参与方式与我们的心理和社会身份如此紧密地交织在一起，那么人们很难想象个人与数学的参与方式也会如此不同。
We need not pursue these questions of essence and accident much further to make the important point, and a point which is widely ignored: If styles of involvement with motorcycle maintenance are so intricately interwoven with our psychological and social identities, one would scarcely expect this to be less true about the varieties of involvements of individuals with mathematics.
本书前面通过海龟几何阐明了这些关于数学工作与整个人的关系的想法，因为它与LOGO编程语言一起使用。这些实验表达了对传统学校数学的批判（这既适用于所谓的新数学，也适用于旧数学）。根据我们在本文中提出的概念来描述传统学校数学，会发现它是数学的非人格化、纯逻辑的“形式化”化身的讽刺画。虽然我们可以证明数学教师的修辞术有所进步（新数学的教师被教导用“理解”和“发现”来说话），但由于他们所教的内容，问题仍然存在。II
These ideas about the relationship of mathematical work with the whole person were illuminated earlier in this book by Turtle geometry, as it is used with the LOGO programming language. These experiments express a critique of traditional school mathematics (which applies no less to the so-called new math than to the old). A description of traditional school mathematics in terms of the concepts we have developed in this essay would reveal it to be a caricature of mathematics in its depersonalized, purely logical, “formal” incarnation. Although we can document progress in the rhetoric of math teachers (teachers of the new math are taught to speak in terms of “understanding” and “discovery”), the problem remains because of what they are teaching.II
在海龟几何中，我们创造了一个环境，在这个环境中，孩子的任务不是学习一套正式的规则，而是对自己在空间中的移动方式有充分的了解，以便将这种自我知识转换成程序，让海龟移动。现在，这本书的读者已经非常熟悉这种控制论动物的潜力。但我想在这里回顾并强调海龟几何的两个密切相关的方面，这两个方面与本文的关注点直接相关。第一是自我共振数学的发展，实际上是“身体共振”数学的发展；第二是数学工作环境的发展，在这个环境中，美学维度（即使是最狭义的“美丽”）也始终被放在首位。
In Turtle geometry we create an environment in which the child’s task is not to learn a set of formal rules but to develop sufficient insight into the way he moves in space to allow the transposition of this self-knowledge into programs that will cause a Turtle to move. By now the reader of this book is very familiar with the potential of this cybernetic animal. But what I would like to do here is recall and underscore two closely related aspects of Turtle geometry which are directly relevant to the concerns of this essay. The first is the development of an ego-syntonic mathematics, indeed, of a “body-syntonic” mathematics; the second is the development of a context for mathematical work where the aesthetic dimension (even in its narrowest sense of “the pretty”) is continually placed in the forefront.
我们将举一个例子来阐明这两个方面：一个典型问题的例子，当一个孩子学习海龟几何时就会出现。孩子已经学会了如何命令海龟朝它面对的方向前进，并绕着它的轴旋转，也就是说，按照孩子命令的度数向右或向左旋转。有了这些命令，孩子就编写了程序，让海龟画出直线图形。孩子迟早会提出这样的问题：“我怎样才能让海龟画一个圆？”在LOGO中，我们不提供“答案”，而是鼓励学习者用自己的身体去寻找解决方案。孩子开始绕圈行走，并发现如何通过向前走一点、转一点、向前走一点、转一点来画一个圆。现在孩子知道如何让海龟画一个圆了：只要给海龟下达与自己相同的命令即可。 “向前走一点，转一点”在龟语中表达为重复[向前1向右 转1]。因此，我们看到了一个既是自我共振又是身体共振的几何推理过程。一旦孩子知道如何以光速在屏幕上放置圆圈，形状、形式和运动的无限调色板就被打开了。因此，发现圆圈（当然还有曲线）是孩子通过数学获得直接审美体验能力的转折点。
We shall give a single example which illuminates both of these aspects: an example of a typical problem that arises when a child is learning Turtle geometry. The child has already learned how to command the Turtle to move forward in the direction that it is facing and to pivot around its axis, that is, to turn the number of degrees right or left that the child has commanded. With these commands the child has written programs which cause the Turtle to draw straight line figures. Sooner or later the child poses the question: “How can I make the Turtle draw a circle?” In LOGO we do not provide “answers,” but encourage learners to use their own bodies to find a solution. The child begins to walk in circles and discovers how to make a circle by going forward a little and turning a little, by going forward a little and turning a little. Now the child knows how to make the Turtle draw a circle: Simply give the Turtle the same commands one would give oneself. Expressing “go forward a little, turn a little” comes out in Turtle language as REPEAT [FORWARD 1 RIGHT TURN 1]. Thus we see a process of geometrical reasoning that is both ego syntonic and body syntonic. And once the child knows how to place circles on the screen with the speed of light, an unlimited palette of shapes, forms, and motion has been opened. Thus the discovery of the circle (and, of course, the curve) is a turning point in the child’s ability to achieve a direct aesthetic experience through mathematics.
上面的段落听起来好像自我协调数学是最近才发明的。事实并非如此，事实上，这与本文反复提出的观点相矛盾，即数学家的数学具有深刻的个人性。我们也没有为孩子们发明自我协调数学。我们只是给孩子们提供了一种重新利用一直属于他们的东西的方法。大多数人觉得他们与数学没有“个人”的联系，然而，作为孩子，他们为自己构建了数学。让·皮亚杰关于发生认识论的研究告诉我们，从生命的第一天起，孩子就从事着从身体与环境的交汇中提取数学知识的事业。关键是，无论我们是否有意为之，数学教学，就像我们学校传统上所做的那样，是一个我们要求孩子忘记数学的自然经验以学习一套新规则的过程。
In the above paragraph it sounds as though ego-syntonic mathematics was recently invented. This is certainly not the case and, indeed, would contradict the point made repeatedly in this essay that the mathematics of the mathematician is profoundly personal. It is also not the case that we have invented ego syntonic mathematics for children. We have merely given children a way to reappropriate what was always theirs. Most people feel that they have no “personal” involvement with mathematics, yet as children they constructed it for themselves. Jean Piaget’s work on genetic epistemology teaches us that from the first days of life a child is engaged in an enterprise of extracting mathematical knowledge from the intersection of body with environment. The point is that, whether we intend it or not, the teaching of mathematics, as it is traditionally done in our schools, is a process by which we ask the child to forget the natural experience of mathematics in order to learn a new set of rules.
直到最近，这种忘记逻辑外根源的过程才在学术界主导了数学的正式历史。在二十世纪早期，形式逻辑被视为数学基础的同义词。直到布尔巴基的结构主义理论出现，我们才看到数学的内部发展，这使得数学能够“记住”其遗传根源。这种“记忆”是为了将数学与关于儿童如何构建现实的研究发展建立最密切的关系。
This same process of forgetting extralogical roots has until very recently dominated the formal history of mathematics in the academy. In the early part of the twentieth century, formal logic was seen as synonymous with the foundation of mathematics. Not until Bourbaki’s structuralist theory appeared do we see an internal development in mathematics which opens mathematics up to “remembering” its genetic roots. This “remembering” was to put mathematics in the closest possible relationship to the development of research about how children construct their reality.
这些思潮以及我们之前在认知和动态心理学中遇到的思潮的影响，将理解数学的事业推向了一个新时代的门槛。沃伦·麦卡洛克的一句警句预示着这一时代：无论是人还是数学，都无法脱离彼此而完全被理解。当被问及什么问题将指导他的科学生涯时，麦卡洛克回答道：“是什么造就了人，使他能够理解数字？又是什么造就了数字，使人能够理解它？”
The consequences of these currents and those we encountered earlier in cognitive and dynamic psychology place the enterprise of understanding mathematics at the threshold of a new period heralded by Warren McCulloch’s epigrammatic assertion that neither man nor mathematics can be fully grasped separately from the other. When asked what question would guide his scientific life, McCulloch answered: “What is a man so made that he can understand number and what is number so made that a man can understand it?”
脚注
Footnotes
我要感谢麻省理工学院出版社的编辑们允许我转载这篇论文，该论文最初发表在 Judith Wechsler 主编的《科学美学》（马萨诸塞州剑桥：麻省理工学院出版社，1978 年）中，题为“庞加莱和数学无意识”。我还要感谢 Judith Wechsler 鼓励我写这篇论文（最初是她在麻省理工学院的一门课上的客座讲座），以及许多其他事情。
I I would like to thank the editors of the MIT press for their permission to reprint this essay which originally appeared as “Poincaré and the Mathematical Unconscious” in Judith Wechsler, ed., Aesthetics in Science (Cambridge, Mass.: MIT Press, 1978). I also want to thank Judith Wechsler for encouraging me to write this essay (which began as a guest lecture in one of her classes at MIT) and for much else as well.
II下列段落已修改以便与本书保持连续性。
II The following paragraphs have been modified for continuity with this book.
后记和致谢
Afterword and Acknowledgments
1964 年，我从一个世界来到了另一个世界。此前五年，我一直住在瑞士日内瓦附近的阿尔卑斯山村庄，与让·皮亚杰一起工作。我关注的焦点是儿童、思考的本质以及儿童如何成为思考者。我搬到麻省理工学院，进入了一个充满控制论和计算机的城市世界。我的注意力仍然集中在思考的本质上，但现在我最关心的是人工智能问题：如何制造会思考的机器？
In 1964 I moved from one world to another. For the previous five years I had lived in Alpine villages near Geneva, Switzerland, where I worked with Jean Piaget. The focus of my attention was on children, on the nature of thinking, and on how children become thinkers. I moved to MIT into an urban world of cybernetics and computers. My attention was still focused on the nature of thinking, but now my immediate concerns were with the problem of artificial intelligence: How to make machines that think?
两个世界截然不同。但我之所以做出转变，是因为我相信我的新机器世界可以提供一个视角，解决我们在旧世界儿童中无法解决的问题。回想起来，我发现这种相互影响带来了双向利益。几年来，马文·明斯基和我一直在研究一种智力的一般理论（称为“心智社会理论”），该理论源于一种同时思考儿童如何思考和计算机如何思考的策略。
Two worlds could hardly be more different. But I made the transition because I believed that my new world of machines could provide a perspective that might lead to solutions to problems that had eluded us in the old world of children. Looking back I see that the cross-fertilization has brought benefits in both directions. For several years now Marvin Minsky and I have been working on a general theory of intelligence (called “The Society Theory of Mind”) which has emerged from a strategy of thinking simultaneously about how children do and how computers might think.
当然，明斯基和我并不是唯一利用计算理论（或信息处理理论）作为解释心理现象的模型来源的学者。相反，沃伦·麦卡洛克、艾伦·纽厄尔、赫伯特·西蒙、阿兰·图灵、诺伯特·维纳以及许多年轻人都采取了这种方法。但本书的出发点是一种观点——与明斯基共同提出——它使我们与公司其他大多数成员截然不同：也就是说，我们不仅将计算机科学的思想视为解释学习和思考如何运作的工具，而且还将其视为可能改变甚至改善人们学习和思考方式的变革工具。
Minsky and I, of course, are not the only workers to have drawn on the theory of computation (or information processing) as a source of models to be used in explaining psychological phenomena. On the contrary, this approach has been taken by such people as Warren McCulloch, Allen Newell, Herbert Simon, Alan Turing, Norbert Wiener, and quite a number of younger people. But the point of departure of this book is a point of view—first articulated jointly with Minsky—that separates us quite sharply from most other members of this company: that is to say, seeing ideas from computer science not only as instruments of explanation of how learning and thinking in fact do work but also as instruments of change that might alter, and possibly improve, the way people learn and think.
这本书源于一个旨在探索这一概念的项目，该项目旨在让孩子们接触“计算机科学的精华”，包括一些最好的技术和一些最好的想法。该项目的核心是在麻省理工学院人工智能实验室和计算机科学实验室（MAC项目）所在的同一栋大楼内创建一个儿童学习环境。我们希望通过将儿童和主要对儿童感兴趣的人带入这个计算机和计算机专家的世界，为思想流入教育创造条件。
The book grew out of a project designed to explore this concept by giving children access to “the best of computer science” including some of its best technology and some of its best ideas. At the heart of the project was the creation of a children’s learning environment in the same building that houses MIT’s Artificial Intelligence Laboratory and Laboratory for Computer Science (Project MAC). We hoped that by bringing children and people interested primarily in children into this world of computers and computerists, we would create conditions for a flow of ideas into thinking about education.
我不会试图描述这个项目过程中发生的一切或从中学到的一切，但我将集中讨论一些个人反思。想要了解更多有关该项目本身的读者可以在书末的注释中找到其他出版物的链接。
I shall not try to describe all that happened in the course of this project or all that was learned from it, but I shall concentrate on some personal reflections. Readers who want to know more about the project itself will find pointers to other publications in the notes at the end of the book.
该项目实际上是一个文化互动实验。它旨在在一个充满某种“计算机文化”的环境中培育一种新的“教育文化”。参与其中的人太多了，我甚至都不知道他们的名字。思想交流更多地发生在午夜后安静的谈话中（因为这是一种不遵守传统时钟周期的计算机文化），而不是在有组织的研讨会或书面论文中进行。在早期的草稿中，我试图记录这种文化的发展。但这太难了，最后我以非常个人化的风格写了这本书。这有一定的优势，让我可以更自由地表达我对其他参与者可能非常不同的看法的想法和事件的个人解释。我希望这不会掩盖我对社区的归属感和表达一套共同思想的感觉。我很遗憾，篇幅不允许我展示其中一些想法是如何被其他人采纳并发展成更先进的形式的。
The project is really an experiment in cultural interaction. It set out to grow a new “education culture” in an environment permeated with a particular form of “computer culture.” Too many people were involved for me even to know all their names. The interchanges of ideas took place much more in conversations in the quiet of after-midnight hours (for this is a computer culture that does not respect the conventional clock cycles) than in organized seminars or written papers. In early drafts I attempted to chronicle the growth of the culture. But it proved too difficult, and in the end I wrote the book in a very personal style. This has a certain advantage in allowing me to give freer reign to my own personal interpretations of ideas and incidents that other participants might well see very differently. I hope that it does not obscure my sense of belonging to a communtity and of expressing a set of shared ideas. I regret that space does not permit me to show how some of these ideas have been picked up by others and elaborated into much more advanced forms.
在本书思想的形成过程中，马文·明斯基是我思想生活中最重要的人。正是从他那里，我第一次了解到计算不仅仅是一门理论科学和一门实用艺术：它还可以成为塑造强大而个性化的世界观的材料。从那时起，我遇到了几个成功做到了这一点的人，他们以鼓舞人心的方式做到了这一点。其中，艾伦·凯脱颖而出，因为他一直将自己的个人计算愿景转向儿童。在整个 20 世纪 70 年代，凯在施乐帕洛阿尔托研究中心的研究小组和我们在麻省理工学院的小组是美国仅有的两位儿童计算机研究人员，他们明确表示，重要的研究不能基于当时在学校、资源中心和教育研究实验室中可用的原始计算机。对我来说，“计算机就像铅笔”这句话唤起了我对未来儿童使用计算机的想象。铅笔既可用于书写，也可用于涂鸦，既可用于画画，既可用于做私人笔记，也可用于完成正式任务。Kay 和我都曾有一个共同的愿景，即计算机将可以更随意、更个人地用于各种用途。但无论是 1970 年的学校计算机终端还是 1980 年的 Radio Shack 家用计算机，都无法提供接近这一愿景的功能和灵活性。为了实现这一目标，计算机必须提供比 1970 年代计算机更好的图形和更灵活的语言，而且价格要让学校和个人能够负担得起。
Marvin Minsky was the most important person in my intellectual life during the growth of the ideas in this book. It was from him that I first learned that computation could be more than a theoretical science and a practical art: It can also be the material from which to fashion a powerful and personal vision of the world. I have since encountered several people who have done this successfully and in an inspirational way. Of these, one who stands out because he has so consistently turned his personal computational vision to thinking about children is Alan Kay. During the whole decade of the 1970s, Kay’s research group at the Xerox Palo Alto Research Center and our group at MIT were the only American workers on computers for children who made a clear decision that significant research could not be based on the primitive computers that were then becoming available in schools, resource centers, and education research laboratories. For me, the phrase “computer as pencil” evokes the kind of uses I imagine children of the future making of computers. Pencils are used for scribbling as well as writing, doodling as well as drawing, for illicit notes as well as for official assignments. Kay and I have shared a vision in which the computer would be used as casually and as personally for an even greater diversity of purposes. But neither the school computer terminal of 1970 nor the Radio Shack home computer of 1980 have the power and flexibility to provide even an approximation of this vision. In order to do so, a computer must offer far better graphics and a far more flexible language than computers of the 1970s can provide at a price schools and individuals can afford.
1967 年，在麻省理工学院的儿童实验室正式成立之前，我开始考虑设计一种适合儿童使用的计算机语言。这并不意味着它应该是一种“玩具”语言。相反，我希望它具有专业编程语言的功能，但我也希望它为非数学初学者提供简单的入门途径。Bolt Beranek and Newman 研究公司教育技术组负责人 Wallace Feurzeig 很快意识到了这个想法的优点，并为该语言的首次实施和试验找到了资金。新语言被命名为LOGO ，以表明它主要是符号性的，其次是定量性的。在与麻省理工学院人工智能小组的首批研究生之一 Daniel Bobrow、当时都在 Bolt Baranek and Newman 工作的 Cynthia Solomon 和 Richard Grant 讨论的过程中，我对该语言的原始设计得到了极大的改进。LOGO语言经过几轮“现代化”改造后，后续大部分开发工作都在麻省理工学院进行。为该语言做出贡献的人很多，我只能列出其中几位：Harold Abelson、Bruce Edwards、Andrea diSessa、Gary Drescher、Ira Goldstein、Mark Gross、Ed Hardeback、Danny Hillis、Bob Lawler、Ron Lebel、Henry Lieberman、Mark Miller、Margaret Minsky、Cynthia Solomon、Wade Williams 和 Terry Winograd。多年来，Ron Lebel 一直担任负责LOGO开发的首席系统程序员。但直接参与LOGO开发的人员只是冰山一角：麻省理工学院社区对LOGO的影响远不止于此。
In 1967, before the children’s laboratory at MIT had been officially formed, I began thinking about designing a computer language that would be suitable for children. This did not mean that it should be a “toy” language. On the contrary, I wanted it to have the power of professional programming languages, but I also wanted it to have easy entry routes for nonmathematical beginners. Wallace Feurzeig, head of the Educational Technology Group at the research firm of Bolt Beranek and Newman, quickly recognized the merit of the idea and found funding for the first implementation and trial of the language. The name LOGO was chosen for the new language to suggest the fact that it is primarily symbolic and only secondarily quantitative. My original design of the language was greatly improved in the course of discussions with Daniel Bobrow, who had been one of the first graduate students in the MIT Artificial Intelligence group, Cynthia Solomon, and Richard Grant, all of whom were working at that time at Bolt Baranek and Newman. Most subsequent development of the LOGO language, which has gone through several rounds of “modernization,” took place at MIT. Of the very many people who contributed to it, I can list only a few: Harold Abelson, Bruce Edwards, Andrea diSessa, Gary Drescher, Ira Goldstein, Mark Gross, Ed Hardeback, Danny Hillis, Bob Lawler, Ron Lebel, Henry Lieberman, Mark Miller, Margaret Minsky, Cynthia Solomon, Wade Williams, and Terry Winograd. For many years Ron Lebel was the chief systems programmer in charge of LOGO development. But the people who worked directly on LOGO form only the tip of an iceberg: The influence of the MIT community on LOGO went much deeper.
我们的人工智能实验室一直处于计算机界强烈反主流运动的中心，这种运动认为编程语言具有强烈的认识论和美学意义。在我看来，这种“沃尔夫式”观点在三位计算机科学家的作品中得到了最好的阐述，他们在LOGO成立时都是研究生：卡尔·休伊特、杰拉德·萨斯曼和特里·维诺格拉德。但它可以追溯到麻省理工学院人工智能小组的创始人马文·明斯基和约翰·麦卡锡，并且很大程度上归功于“黑客”的传统，我最直接感受到威廉·高斯珀和理查德·格林布拉特的影响。在这些人创造的文化氛围中，让孩子们通过学习BASIC等计算机语言进入计算机文化是不可接受的，就像让他们接触英语诗歌仅限于洋泾浜英语翻译一样不可接受。
Our Artificial Intelligence Laboratory has always been near the center of a movement, strongly countercultural in the larger world of computers, that sees programming languages as heavily invested with epistemological and aesthetic commitments. For me this “Whorfian” view has been best articulated in the work of three computer scientists who were graduate students at the time LOGO was in formation: Carl Hewitt, Gerald Sussman, and Terry Winograd. But it goes back to the founders of the MIT Artificial Intelligence group, Marvin Minsky and John McCarthy, and owes much to the tradition of “hackers” of whom I feel most directly the influence of William Gosper and Richard Greenblatt. In the cultural atmosphere created by such people it was as unacceptable for children to enter the computer culture by learning computer languages such as BASIC as it would be to confine their access to English poetry to pidgin English translations.
我一直认为学习是一种爱好，通过培养对如何做这件事的敏感性，我对它的本质有了许多深刻的见解。因此，我可能比大多数人更刻意地学习了更广泛的材料。我本着这种精神学到的东西包括科学章节（如热力学）、阅读汉字、驾驶飞机、烹饪各种菜肴、表演杂耍等马戏艺术，甚至两次戴着扭曲眼镜生活了几个星期（一次是左右翻转眼镜，另一次是相当复杂的视野棱镜扭曲）。我发现人工智能社区如此吸引人的部分原因是，人们对这种将自己作为洞察心理过程的来源的方法有着共同的兴趣，并且特别喜欢观察自己从事熟练的活动。在这里，我再次感谢许多人，我只能挑出那些贡献最为突出的人：霍华德·奥斯汀、珍妮·班伯格、艾拉·戈尔茨坦、鲍勃·劳勒、杰拉尔德·萨斯曼，以及参加我的“大声思考研讨会”的研究生，在研讨会上探讨了这种方法。在与唐纳德·肖恩和本森·斯奈德合作期间，以及与伊迪丝·阿克曼、丹尼尔·鲍勃罗、霍华德·格鲁伯、安妮特·卡米洛夫-史密斯和唐纳德·诺曼等多位心理学家的互动中，我对“大声思考”的方法变得更加成熟。
I have always considered learning a hobby and have developed many insights into its nature by cultivating a sensitivity to how I go about doing it. Thus, I have perhaps engaged in deliberate learning of a wider range of material than most people. Examples of things I have learned in this spirit include chapters of science (like thermodynamics), reading Chinese characters, flying airplanes, cooking in various cuisines, performing circus arts such as juggling, and even two bouts of living for several weeks with distorting spectacles (on one occasion left-right reversing glasses, on the other a rather complex prismatic distortion of the visual field). Part of what I found so attractive about the artificial intelligence community was a shared interest in this approach to using one’s self as a source of insight into psychological processes and a particular interest in observing oneself engaged in skilled activities. Here again I owe debts to many people and am able to single out only those whose contributions were most salient: Howard Austin, Jeanne Bamberger, Ira Goldstein, Bob Lawler, Gerald Sussman, and the graduate students who took part in my “loud thinking seminars” where such methods were explored. My approach to “loud thinking” acquired greater sophistication during a period of collaboration with Donald Schon and Benson Snyder and in interaction with a number of psychologists including Edith Ackermann, Daniel Bobrow, Howard Gruber, Annette Karmiloff-Smith, and Donald Norman.
所有这些影响都促成了我们为儿童构建的计算环境中学习/教学方法的出现。在这项工作中，与我最亲近的人是辛西娅·所罗门。与马文·明斯基一样，我与她的合作非常密切，持续了很长时间，以至于我无法一一列举她所做的重大贡献。所罗门也是第一个开发出一种智力连贯的方法，用于培训教师向儿童介绍计算机的人，并且仍然是少数以应有的严肃态度处理这个问题的人之一。
All these influences entered into the emergence of a learning/teaching methodology in the computational environments we were building for children. The person closest to me in this work was Cynthia Solomon. As in the case of Marvin Minsky, my collaboration with her was so close over so long a period that I find it impossible to enumerate the substantial contributions she made. Solomon was also the first to develop an intellectually coherent methodology for training teachers to introduce children to computers and is still one of the few people to have approached this problem with the seriousness it deserves.
许多人都对如何教孩子LOGO贡献了自己的想法。Ira Goldstein 承担了开发教学过程理论框架这一难题，Mark Miller 紧随其后。其他人则以更务实的精神进行教学。Howard Austin、Paul Goldenberg、Gerianne Goldstein、Virginia Grammar、Andree Green、Ellen Hildreth、Kiyoko Okumura、Neil Rowe 和 Dan Watt 做出了特别贡献。Jeanne Bamberger 开发了在音乐学习中使用LOGO的方法，并提高了教师对自己思维的敏感性。
Many people contributed ideas about teaching children LOGO. Ira Goldstein undertook the difficult problem of developing a theoretical framework for the instructional process and was followed in this work by Mark Miller. Others approached teaching in a more pragmatic spirit. Special contributions have been made by Howard Austin, Paul Goldenberg, Gerianne Goldstein, Virginia Grammar, Andree Green, Ellen Hildreth, Kiyoko Okumura, Neil Rowe, and Dan Watt. Jeanne Bamberger developed methods for using LOGO in musical learning and in increasing teachers’ sensitivity to their own thinking.
我们学习环境背后的一个核心思想是，孩子们能够利用数学和科学中的强大思想作为个人力量的工具。例如，几何学将成为在电视屏幕上创造视觉效果的一种手段。但要实现这一点，通常意味着开发数学和科学的新课题，而这一事业之所以能够实现，完全是因为我们在一个拥有大量富有创造力的数学人才的机构内工作。这项任务是一种新任务：它包括进行数学或科学方面的真正原创研究，但选择的方向是因为它们可以带来更易理解或更易学习的知识形式，而不是出于通常激发数学研究的那些原因。麻省理工学院的许多学生和教职员工都参与了这项工作，但有两位是该领域的专业人士：数学家 Harold Abelson 和物理学家 Andrew diSessa。
A central idea behind our learning environments was that children would be able to use powerful ideas from mathematics and science as instruments of personal power. For example, geometry would become a means to create visual effects on a television screen. But achieving this often meant developing new topics in mathematics and science, an enterprise that was possible only because we were working within an institution rich in creative mathematical talent. The task is of a new kind: It consists of doing what is really original research in mathematics or science but in directions chosen because they lead to more comprehensible or more learnable forms of knowledge and not for the kinds of reasons that typically motivate mathematical research. Many students and faculty members at MIT contributed to this work, but two stand out as professionals in the area: Harold Abelson, a mathematician, and Andrew diSessa, a physicist.
许多LOGO工作人员为 Turtle 图画的美感做出了贡献。对我影响最大的是 Cynthia Solomon、Ellen Hildreth 和 Ilse Schenck（她为本书设计了花园和鸟类）。
Many LOGO workers contributed to the aesthetic of the Turtle drawings. Those who most influenced me were Cynthia Solomon, Ellen Hildreth, and Ilse Schenck (who arranged the garden and birds in this book).
在这本书中，我写的是儿童，但事实上，书中表达的大多数观点都与人们在任何年龄的学习方式有关。我特别提到儿童，以反映我个人的信念：学习条件的变化最能受益的是最小的孩子。与我们合作的大多数孩子都是小学中学生。Radia Perlman 是第一个探索与年龄小得多的孩子（最小的只有四岁）一起工作的技术的人。Abelson 和 diSessa 专门研究高中和大学年龄的年龄较大的学生。Gary Drescher、Paul Goldenberg、Sylvia Weir 和 Jose Valente 是向严重残疾儿童教授LOGO 的先驱。Bob Lawler 进行了第一个也是迄今为止唯一一个不同类型的学习实验，我认为这种实验在未来将变得非常重要。在劳勒的研究中，对一个孩子进行了为期六个月的“全程”观察，以便不仅捕捉在人为情况下发生的学习，而且捕捉在此期间发生的所有公开学习。我还受到了另一项关于“自然学习”的研究的影响，该研究是劳伦斯·米勒在哈佛大学论文研究的一部分。劳勒和米勒都为本书所基于的一般智力立场提供了数据：最好的学习发生在学习者掌控的时候。埃德温娜·米奇纳的博士论文是一项非常不同的学习研究，试图描述数学文化没有写在书中的一些数学知识。
In this book I write about children but, in fact, most of the ideas expressed are relevant to how people learn at any age. I make specific references to children as a reflection of my personal conviction that it is the very youngest who stand to gain the most from change in the conditions of learning. Most of the children who collaborated with us were of mid-elementary school age. Radia Perlman was the first to explore techniques for working with much younger children, as young as four years of age. Abelson and diSessa have specialized in work with older students of high school and college age. Gary Drescher, Paul Goldenberg, Sylvia Weir, and Jose Valente are among those who have pioneered teaching LOGO to severely handicapped children. Bob Lawler carried out the first, and so far the only, example of a different kind of learning experiment, a kind that I think will become very important in the future. In Lawler’s study, a child was observed “full time” during a six-month period so as to capture not only the learning that took place in contrived situations but all the overt learning that took place during that period. I have also been influenced by another study on “natural learning” now being conducted as part of research by Lawrence Miller for his thesis at Harvard. Both Lawler and Miller provided data for a general intellectual position that underlies this book: The best learning takes place when the learner takes charge. Edwina Michner’s Ph.D. thesis was a learning study of a very different sort, an attempt to characterize some of the mathematical knowledge that the mathematical culture does not write down in its books.
我承认许多人都曾对我负有智力上的义务。我还要感谢他们中的大多数人，感谢他们给予我的支持和对我经常混乱的工作方式的耐心。我非常感谢所有容忍我的人，尤其是 Gregory Gargarian，他承担了非常艰巨的工作，维护了LOGO实验室的组织，并在计算机文件中输入和更新了本书的许多连续版本。除了他的能力和专业精神之外，他的友谊和支持也使本书的写作过程变得轻松了许多。
I have acknowledged intellectual obligations to many people. I have to thank most of them for something else as well: for support and for patience with my too often disorganized working style. I am deeply grateful to everyone who put up with me, especially Gregory Gargarian who had the very difficult jobs of maintaining the organization of the LOGO Laboratory and of entering and updating many successive versions of this book in the computer files. In addition to his competence and professionalism, his friendship and support have made easier many moments in the writing of this book.
MIT提供了极具启发性的智力环境。其行政环境也非常特殊，允许不同寻常的项目蓬勃发展。许多人在行政方面提供了帮助：杰罗姆·威斯纳 (Jerome Wiesner)、沃尔特·罗森布利思 (Walter Rosenblith)、迈克尔·德图佐斯 (Michael Dertouzos)、泰德·马丁 (Ted Martin)、本森·斯奈德 (Benson Snyder)、帕特里克·温斯顿 (Patrick Winston)、芭芭拉·纳尔逊 (Barbara Nelson)、伊娃·坎皮茨 (Eva Kampits)、吉姆·麦卡锡 (Jim McCarthy)、戈登·奥罗 (Gordon Oro) 和乔治·华莱士 (George Wallace)，但我相信还有很多其他人。其中，我特别要感谢伊娃·坎皮茨 (Eva Kampits)，她曾是我的秘书，现在是坎皮茨博士。
MIT has provided a highly stimulating intellectual environment. Its administrative environment is also very special in allowing out-of-the-ordinary projects to flourish. Many people have helped in an administrative capacity: Jerome Wiesner, Walter Rosenblith, Michael Dertouzos, Ted Martin, Benson Snyder, Patrick Winston, Barbara Nelson, Eva Kampits, Jim McCarthy, Gordon Oro, and George Wallace come to mind but I am sure there are many others. Of these I owe a very special debt to Eva Kampits who was once my secretary and is now Dr. Kampits.
如果没有我之前提到的不同类型的支持， LOGO项目就不可能实现。美国国家科学基金会自LOGO成立之初就一直支持该项目。我还想提到一些基金会的个人，他们的富有想象力的理解使我们得以开展工作：Dorothy Derringer、Andrew Molnar 和 Milton Rose。这些人给予的支持既有精神价值，也有物质价值，我想将福特基金会的 Marjorie Martus、美国国家教育研究所的 Arthur Melmed、残疾人教育局的 Alan Ditman 和德州仪器的 Alfred Riccomi 也包括在内。我还特别要提到三位给予我们精神和物质支持的个人：Ida Green、Erik Jonsson 和 Cecil Green，他们都来自德克萨斯州的达拉斯。与 Erik Jonsson 密切合作，在达拉斯的 Lamplighter 学校使用计算机开发项目，对我来说是一次特别丰富的经历。我渐渐欣赏他清晰的思路和广阔的视野，把他视为同事和朋友。他对我的想法的支持和对我杂乱无章的容忍帮助我创作了这本书。
The LOGO project could not have happened without support of a different kind than I have mentioned until now. The National Science Foundation has supported the work on LOGO since its inception. I want also to mention some of the Foundation’s individuals whose imaginative understanding made it possible for us to do our work: Dorothy Derringer, Andrew Molnar, and Milton Rose. The value of the support given by such people is moral as well as material, and I would include in this category Marjorie Martus at the Ford Foundation, Arthur Melmed at the National Institute of Education, Alan Ditman at the Bureau for the Education of the Handicapped, and Alfred Riccomi of Texas Instruments. I would also most especially include three individuals who have given us moral and material support: Ida Green, Erik Jonsson, and Cecil Green all from Dallas, Texas. It has been a particularly rich experience for me to work closely with Erik Jonsson on developing a project using computers in the Lamplighter School in Dallas. I have come to appreciate his clarity of thought and breadth of vision and to think of him as a colleague and a friend. His support for my ideas and intolerance of my disorganization helped make this book happen.
约翰·伯洛对本书的写作贡献巨大。他是一位非常聪明的编辑。在手稿编写的每个阶段，他批判性和热情的阅读都带来了新的清晰度和新的想法。随着项目的发展，对我来说，他不仅仅是一名编辑。他成为了我的朋友、对话伙伴、评论家，以及我最想影响的读者类型的典范。当我遇到约翰时，他没有计算机专业知识，但他在其他领域的知识为他提供了一个直接的基础，使他能够从中产生自己关于计算机和教育的想法。
John Berlow contributed beyond measure to the writing of this book. He came into the picture as an unusually intelligent editor. At every phase in the manuscript’s development, his critical and enthusiastic readings led to new clarity and new ideas. As the project developed he became, for me, more than an editor. He became a friend, a dialog partner, a critic, and a model of the kind of reader I most want to influence. When I met John he was without computer expertise, although his knowledge in other areas provided him with an immediate base from which to generate his own ideas concerning computers and education.
许多人的贡献是无法归类的。尼古拉斯·尼葛洛庞帝 (Nicholas Negroponte) 是源源不断的灵感源泉，部分原因正是因为他无法被归类。我还要感谢苏珊·哈特内特 (Susan Hartnett)、安德鲁·恩里克斯 (Androula Henriques)、巴贝尔·因赫尔德 (Barbel Inhelder)、AR·琼克希 (AR Jonckheere)、邓肯·斯图尔特·林尼 (Duncan Stuart Linney)、艾伦·帕普特 (Alan Papert)、多纳·施特劳斯 (Dona Strauss) 和 IB Tabata。还有一些人，他们对于如何使用计算机的分歧一直很有价值：约翰·西利·布朗 (John Seeley Brown)、艾拉·戈尔茨坦 (Ira Goldstein)、罗伯特·戴维斯 (Robert Davis)、亚瑟·勒尔曼 (Arthur Leuhrman)、帕特里克·苏佩斯 (Patrick Suppes)。如果这本书可以看作是积极乐观思想的表达，那么这必须归功于我的母亲贝蒂·帕普特 (Betty Papert)。阿特米斯·帕普特 (Artemis Papert) 在许多方面都给予了我很大的帮助，我只能说：谢谢。
There are many people whose contributions cannot be categorized. Nicholas Negroponte is a constant source of inspiration, in part precisely because he defies categorization. I also wish to thank Susan Hartnett, Androula Henriques, Barbel Inhelder, A. R. Jonckheere, Duncan Stuart Linney, Alan Papert, Dona Strauss, and I. B. Tabata. And there are a few people with whom disagreements about how computers should be used have always been valuable: John Seeley Brown, Ira Goldstein, Robert Davis, Arthur Leuhrman, Patrick Suppes. If the book can be read as an expression of positive and optimistic thinking, this must be attributed to my mother, Betty Papert. Artemis Papert has helped in so many ways that I can only say: Merci.
所有关心儿童思维方式的人都深受让·皮亚杰的影响。我尤其感激他。如果皮亚杰没有介入我的生活，我现在就会成为一名“真正的数学家”，而不是现在这个样子。皮亚杰在我身上投入了大量精力和信心。我希望他能认识到，我对儿童世界的贡献符合他一生的事业精神。
Everyone concerned with how children think has an immense general debt to Jean Piaget. I have a special debt as well. If Piaget had not intervened in my life I would now be a “real mathematician” instead of being whatever it is that I have become. Piaget invested a lot of energy and a lot of faith in me. I hope that he will recognize what I have contributed to the world of children as being in the spirit of his life enterprise.
离开日内瓦时，我深受皮亚杰儿童形象的启发，尤其是他提出的儿童无需学习就能学到很多东西的想法。但我也非常沮丧，因为他很少告诉我们如何通过这种奇妙的“皮亚杰学习”过程为儿童创造条件，让他们获得更多的知识。我认为设计“皮亚杰课程”的流行想法是对皮亚杰的颠覆：皮亚杰是无课程学习的杰出理论家。因此，我开始构思贯穿本书的两个思想：（1）智力发展模式的重大变化将通过文化变革实现，（2）在不久的将来，最有可能带来潜在相关文化变革的是日益普及的计算机。尽管这些观点从一开始就影响了LOGO项目，但很长一段时间我都不知道如何为它们提供一个理论框架。
I left Geneva enormously inspired by Piaget’s image of the child, particularly by his idea that children learn so much without being taught. But I was also enormously frustrated by how little he could tell us about how to create conditions for more knowledge to be acquired by children through this marvelous process of “Piagetian learning.” I saw the popular idea of designing a “Piagetian Curriculum” as standing Piaget on his head: Piaget is par excellence the theorist of learning without curriculum. As a consequence, I began to formulate two ideas that run through this book: (1) significant change in patterns of intellectual development will come about through cultural change, and (2) the most likely bearer of potentially relevant cultural change in the near future is the increasingly pervasive computer presence. Although these perspectives had informed the LOGO project from its inception, for a long time I could not see how to give them a theoretical framework.
在这方面，以及在许多其他方面，我的妻子雪莉·特克尔都给予了我很大的帮助。没有她，这本书就写不出来。从雪莉那里借来的思想，却成了我试图发展关于计算机和文化的思维方式时缺失的环节。雪莉是一位社会学家，她特别关注思想和文化形成的相互作用，特别是思想复合体是如何被文化群体所采纳和表达的。当我遇到她时，她刚刚完成了一项关于法国新精神分析文化的研究，研究了精神分析如何“殖民”了法国，而法国是一个强烈抵制弗洛伊德影响的国家。她把注意力转向了计算机文化，思考人们与计算的关系如何影响他们的语言、他们对政治的看法以及他们对自己的看法。听她谈论这两个项目，帮助我形成了自己的方法，并让我的想法有了足够的结束感，可以开始这个写作项目。
I was helped in this, as in many other ways, by my wife Sherry Turkle. Without her, this book could not have been written. Ideas borrowed from Sherry turned out to be missing links in my attempts to develop ways of thinking about computers and cultures. Sherry is a sociologist whose particular concerns center on the interaction of ideas and culture formation, in particular how complexes of ideas are adopted by and articulated throughout cultural groups. When I met her she had recently completed an investigation of a new French psychoanalytic culture, of how psychoanalysis had “colonized” France, a country that had fiercely resisted Freudian influence. She had turned her attention to computer cultures and was thinking about how people’s relationships with computation influence their language, their ideas about politics, and their views of themselves. Listening to her talk about both projects helped me to formulate my own approach and to achieve a sufficient sense of closure in my ideas to embark on this writing project.
多年来，雪莉一直给予我各种支持。当写作不顺利时，她会花上几个小时与我交谈，并提供编辑帮助。但当我不再喜欢这本书，或者当我对写这本书的决心失去信心时，她的支持是最决定性的。然后，她对这个项目的承诺使它得以继续，她对我的爱帮助我重新爱上这项工作。
Over the years Sherry has given me every kind of support. When the writing would not work out she gave me hours of conversation and editorial help. But her support was most decisive on the many occasions when I fell out of love with the book or when my confidence in my resolution to write it flagged. Then, her commitment to the project kept it alive and her love for me helped me find my way back to being in love with the work.
西摩·帕珀特
SEYMOUR PAPERT
马萨诸塞州剑桥
Cambridge, Massachusetts
1980 年 4 月
April 1980
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西摩·派普特（1928-2016）曾任麻省理工学院数学与教育学教授，并在那里共同创立了人工智能与媒体实验室。他是LOGO编程语言的共同发明者。他著有《感知器》、《互联 家庭》和《儿童机器》等多部著作。
Seymour Papert (1928-2016) was a professor of mathematics and education at MIT, where he co-founded the Artificial Intelligence and Media Laboratories. He was the co-inventor of the LOGO programming language. He is the author of numerous books, including Perceptrons, The Connected Family, and The Children’s Machine.
米切尔·雷斯尼克 (Mitchel Resnick)是麻省理工学院 ( MIT ) 媒体实验室的乐高帕普特学习研究教授、大川中心主任和终身幼儿园小组主任。雷斯尼克曾是帕普特的学生，他开发了 Scratch，一种旨在帮助儿童学习编码的编程语言，他的项目可编程积木为乐高头脑风暴奠定了基础。他是哈罗德·W·麦格劳 Jr. 教育奖的获得者。雷斯尼克也是《终身幼儿园》一书的作者。
Mitchel Resnick is Lego Papert Professor of Learning Research, Director of the Okawa Center, and Director of the Lifelong Kindergarten group at the Massachusetts Institute of Technology (MIT) Media Lab. A former student of Papert’s, Resnick developed Scratch, a programming language designed to help children learn coding, and his project Programmable Bricks served as the foundation for LEGO Mindstorms. He is the recipient of the Harold W. McGraw Jr. Prize in Education. Resnick is also the author of Lifelong Kindergarten.
笔记
Notes
介绍
INTRODUCTION
1.本书关注的焦点是皮亚杰。我对他的理论立场做了一个略微非正统的解释，对他的理论对教育的影响做了一个非常非正统的解释。想要回到原点的读者需要一些指导，因为皮亚杰写了大量书籍，其中大部分都讨论了儿童发展的特定方面，假设其他书籍是作为理论序言阅读的。关于皮亚杰的最佳短篇书籍是 M. Boden 的《皮亚杰》（伦敦：Harvester Press，1979 年）。阅读皮亚杰本人文本的一个好起点是 HE Gruber 和 JJ Voneche 编辑的《皮亚杰精要：解释性参考和指南》（纽约：Basic Books，1977 年）。我自己列出的皮亚杰最可读且对他的思想提供最佳哲学概述的书籍“简短名单”是：《儿童对世界的概念》（纽约：哈考特，布雷斯公司，1929 年）；《儿童对物理因果关系的概念》（纽约：哈考特，布雷斯公司，1932 年）；《智力心理学》，马尔科姆·皮尔西和 DE·伯林译（纽约：哈考特，布雷斯公司，1950 年）； 《儿童智力的起源》，玛格丽特·库克译（伦敦：劳特利奇和基根·保罗出版社）；《遗传认识论导论》（巴黎：法国大学出版社，1950 年）；《哲学中的洞察与幻觉》，沃尔夫·梅斯译（纽约：世界出版公司，1971 年）；《意识的掌握》，玛格丽特·库克译（伦敦：劳特利奇和基根·保罗出版社）； Susan Wedgwood (剑桥：哈佛大学出版社，1976 年)。关于对“皮亚杰课程开发者”的批评，我曾说他们是“颠倒了皮亚杰”，请参阅 G. Groen，《皮亚杰的理论思想和教育实践》，《研究对教育的影响》，编辑 P. Suppes (华盛顿特区：美国国家教育学院，1978 年)。
1. Piaget is at the center of the concerns of this book. I make a slightly unorthodox interpretation of his theoretical position and a very unorthodox interpretation of the implications of his theory for education. The reader who would like to return to the source needs some guidance because Piaget has written a large number of books, most of which discuss particular aspects of children’s development, assuming that the others have been read as a theoretical preface. The best short book about Piaget is M. Boden’s Piaget (London: Harvester Press, 1979). A good starting place for reading Piaget’s own texts is with H. E. Gruber and J. J. Voneche, eds., The Essential Piaget: An Interpretive Reference and Guide (New York: Basic Books, 1977). My own “short list” of books by Piaget that are most readable and provide the best philosophical overview of his ideas are: The Child’s Conception of the World (New York: Harcourt, Brace and Co., 1929); The Child’s Conception of Physical Causality (New York: Harcourt, Brace and Co., 1932); The Psychology of Intelligence, trans. Malcolm Piercy and D. E. Berlyne (New York: Harcourt, Brace and Co., 1950); The Origins of Intelligence in Children, trans. Margaret Cook (London: Routledge and Kegan Paul); Introduction à l’Epistémologie Génétique (Paris: Presses Universitaires de France, 1950); Insights and Illusions in Philosophy, trans. Wolfe Mays (New York: The World Publishing Co., 1971); The Grasp of Consciousness, trans. Susan Wedgwood (Cambridge: Harvard University Press, 1976). For a critique of the “Piaget Curriculum Developers,” of whom I have said that they are “standing Piaget on his head,” see G. Groen, “The Theoretical Ideas of Piaget and Educational Practice,” The Impact of Research on Education, ed. P. Suppes (Washington D. C.: The National Academy of Education, 1978).
2. LOGO是与之相伴而生的计算机语言家族中一种教育哲学的名称。LOGO语言家族的典型特征包括具有局部变量的程序定义，以允许递归。因此，在LOGO中可以定义新的命令和函数，然后可以像原始命令和函数一样使用它们。LOGO是一种解释性语言。这意味着它可以交互使用。现代LOGO系统具有完整的列表结构，也就是说，该语言可以对列表进行操作，这些列表的成员本身可以是列表、列表的列表等等。
2. LOGO is the name of a philosophy of education in a growing family of computer languages that goes with it. Characteristic features of the LOGO family of languages include procedural definitions with local variables to permit recursion. Thus, in LOGO it is possible to define new commands and functions which then can be used exactly like primitive ones. LOGO is an interpretive language. This means that it can be used interactively. The modern LOGO systems have full list structure, that is to say, the language can operate on lists whose members can themselves be lists, lists of lists, and so forth.
某些版本具有并行处理和消息传递元素，以便于图形编程。列表结构的一个强大用途示例是将LOGO程序本身表示为列表的列表，以便LOGO程序可以构建、修改和运行其他LOGO程序。因此，LOGO不是“玩具”，不是一种只适合儿童的语言。然而，本书中LOGO的简单用法示例确实说明了LOGO的一些特殊之处，因为它旨在为没有数学知识的初学者提供非常早期和简单的编程入门途径。包含 Turtle 命令的LOGO子集（初学者最常用的“入门途径”）在本书中被称为“TURTLE TALK” ，以考虑到其他计算机语言（例如SMALLTALK和PASCAL ）已经使用最初在LOGO语言中开发的命令在其系统上实现了 Turtle。LOGO的 TURTLE TALK 子集可以轻松移植到其他语言中。
Some versions have elements of parallel processing and of message passing in order to facilitate graphics programming. An example of a powerful use of list structure is the representation of LOGO procedures themselves as lists of lists so that LOGO procedures can construct, modify, and run other LOGO procedures. Thus LOGO is not a “toy,” a language only for children. The examples of simple uses of LOGO in this book do however illustrate some ways in which LOGO is special in that it is designed to provide very early and easy entry routes into programming for beginners with no prior mathematical knowledge. The subset of LOGO containing Turtle commands, the most used “entry route” for beginners, is referred to in this book as “TURTLE TALK” to take account of the fact that other computer languages, for example SMALLTALK and PASCAL, have implemented Turtles on their systems using commands originally developed in the LOGO language. The TURTLE TALK subset of LOGO is easily transportable to other languages.
需要牢记的是，LOGO从未被视为最终产品或作为“权威语言”。在这里，我将其作为示例展示，以表明更好的东西是可能的。
It should be carefully remembered that LOGO is never conceived as a final product or offered as “the definitive language.” Here I present it as a sample to show that something better is possible.
正因为LOGO不是玩具，而是一种功能强大的计算机语言，它需要的内存比BASIC等功能较弱的语言大得多。这意味着直到最近，LOGO才只能在相对较大的计算机上实现。随着内存成本的降低，这种情况正在迅速改变。在本书付印时，LOGO系统的原型正在 48 K Apple II 系统和具有扩展内存的TI 99/4 上运行。请参阅 S. Papert 等人的《LOGO ：一种学习语言》（新泽西州莫里斯敦：创意计算出版社，1980 年夏季）。
Precisely because LOGO is not a toy, but a powerful computer language, it requires considerably larger memory than less powerful languages such as BASIC. This has meant that until recently LOGO was only to be implemented on relatively large computers. With the lowering cost of memory, this situation is rapidly changing. As this book goes to press, prototypes of LOGO systems are running on a 48K Apple II system and on a TI 99/4 with extended memory. See S. Papert et al., LOGO: A Language For Learning (Morristown, N.J.: Creative Computing Press, Summer 1980).
3. LOGO项目中 Turtle 的历史如下。1968-1969 年，马萨诸塞州列克星敦 Muzzy 初中的第一批 12 名“普通”七年级学生在整个学年中都使用LOGO来代替他们正常的数学课程。当时LOGO系统没有图形。学生们编写了可以将英语翻译成“Pig Latin”的程序、可以玩策略游戏的程序以及生成具体诗歌的程序。这首次证实了LOGO是一种计算机“新手”可以学习的语言。然而，我希望看到演示扩展到五年级、三年级，最终扩展到学龄前儿童。显然，即使LOGO语言在这些年龄段可以学习，编程主题也不会。我提出 Turtle 是一个可以引起所有年龄段人们兴趣的编程领域。这种期望后来得到了经验的证实，Turtle 作为一种学习设备已被广泛采用。使用海龟编程来教幼儿的先驱者是 Radia Perlman，她在麻省理工学院读书时演示了四岁小孩也能学会控制机械海龟。Cynthia Solomon 首次使用屏幕海龟演示了如何学习一年级学生编程。在另一个年龄段，令人鼓舞的是，大学里开始使用海龟编程来教授PASCAL。请参阅 Kenneth L. Bowles 的《使用PASCAL解决问题》 （纽约：Springer-Verlag，1977 年）。事实证明，控制海龟是一项非常有趣的活动，它对有特殊需要的儿童、自闭症儿童以及患有各种“学习障碍”的儿童都很有吸引力。例如，请参阅 Paul Goldenberg 的《特殊儿童的特殊技术》 （巴尔的摩：大学公园出版社，1979 年）。海龟已经被纳入施乐帕洛阿尔托研究中心的SMALLTALK计算机系统。参见 Alan Kay 和 Adele Goldberg 的《个人动态媒体》（加利福尼亚州帕洛阿尔托：施乐公司，帕洛阿尔托研究中心，1976 年）。
3. The history of the Turtle in the LOGO project is as follows. In 1968–1969, the first class of twelve “average” seventh-grade students at the Muzzy Junior High School in Lexington, Massachusetts, worked with LOGO through the whole school year in place of their normal mathematics curriculum. At that time the LOGO system had no graphics. The students wrote programs that could translate English to “Pig Latin,” programs that could play games of strategy, and programs to generate concrete poetry. This was the first confirmation that LOGO was a learnable language for computer “novices.” However, I wanted to see the demonstration extended to fifth graders, third graders, and ultimately to preschool children. It seemed obvious that even if the LOGO language was learnable at these ages, the programming topics would not be. I proposed the Turtle as a programming domain that could be interesting to people at all ages. This expectation has subsequently been borne out by experience, and the Turtle as a learning device has been widely adopted. Pioneer work in using the Turtle to teach very young children was done by Radia Perlman who demonstrated, while she was a student at MIT, that four-year-old children could learn to control mechanical Turtles. Cynthia Solomon used screen Turtles in the first demonstration that first graders could learn to program. At the other end of the age spectrum, it is encouraging to see that Turtle programming is being used at a college level to teach PASCAL. See Kenneth L. Bowles, Problem Solving Using PASCAL (New York: Springer-Verlag, 1977). Controlling Turtles has proven to be an engaging activity for special needs children, for autistic children, and for children with a variety of “learning disorders.” See for example, Paul Goldenberg, Special Technology for Special Children (Baltimore: University Park Press, 1979). Turtles have been incorporated into the SMALLTALK computer system at the Xerox Palo Alto Research Center. See Alan Kay and Adele Goldberg, “Personal Dynamic Media” (Palo Alto, Calif.: Xerox, Palo Alto Research Center, 1976).
4.触摸传感器 海龟。LOGO中最简单的触摸传感器程序如下：
4. Touch Sensor Turtle. The simplest touch sensor program in LOGO is as follows:
反弹
TO BOUNCE
重复
REPEAT
前进 1
FORWARD 1
测试正面触摸
TEST FRONT TOUCH
如果正确 180
IFTRUE RIGHT 180
结尾
END
评论
Comments
这意味着重复所有单独的步骤
This means repeat all the individual steps
检查是否遇到了什么
It checks whether it has run into something
如果是这样，它就会掉头
If so, it does an about turn
这将使 Turtle 在遇到物体时转向。使用触摸传感器 Turtle 的更微妙且更有指导意义的程序如下：
This will make the Turtle turn about when it encounters an object. A more subtle and more instructive program using the Touch Sensor Turtle is as follows:
重复
REPEAT
前进 1
FORWARD 1
测试左触摸
TEST LEFT TOUCH
如果正确 1
IFTRUE RIGHT 1
假左 1
IFFALSE LEFT 1
结尾
END
评论
Comments
检查：它感人吗？
Check: Is it touching?
它觉得太近了，就转身走开了
It thinks it’s too close and turns away
它以为可能会失去这个物体，所以它转向
It thinks it might lose the object so it turns toward
这个程序将使 Turtle 绕着任意形状的物体飞行，只要它从左侧接触该物体开始（并且该物体及其轮廓中的任何不规则之处与 Turtle 相比都很大）。
This program will cause the Turtle to circumnavigate an object of any shape, provided that it starts with its left side in contact with the object (and provided that the object and any irregularities in its contour are large compared to the Turtle).
对于一组学生来说，这是一个非常有启发性的项目，他们通过演示他们认为如何利用触觉来绕过物体，并将他们的策略转化为海龟命令，从第一原理开发这个（或同等的）程序。
It is a very instructive project for a group of students to develop this (or an equivalent) program from first principles by acting out how they think they would use touch to get around an object and by translating their strategies into Turtle commands.
第一章
CHAPTER 1
1. FOLLOW程序（见引言，注 4）是一个非常简单的例子，它展示了如何利用强大的控制论思想（负反馈控制）来阐明生物或心理现象。这个例子虽然简单，但有助于弥合“因果机制”的物理模型与“目的”等心理现象之间的差距。
1. The program FOLLOW (See Introduction, note 4) is a very simple example of how a powerful cybernetic idea (control by negative feedback) can be used to elucidate a biological or psychological phenomenon. Simple as it is, the example helps bridge the gap between physical models of “causal mechanism” and psychological phenomenon such as “purpose.”
理论心理学家本着同样的精神，使用更复杂的程序构建了几乎所有已知心理现象的模型。赫伯特·A·西蒙 (Herbert A. Simon) 的《人工智能科学》(Sciences of the Artificial ) （剑桥：麻省理工学院出版社，1969 年）大胆阐述了这种探究精神。
Theoretical psychologists have used more complex programs in the same spirit to construct models of practically every known psychological phenomenon. A bold formulation of the spirit of such inquiry is found in Herbert A. Simon, Sciences of the Artificial (Cambridge: MIT Press, 1969).
2.这里提到的批评者和怀疑论者是多年公开和私下辩论的结晶。这些态度被广泛持有，但不幸的是，很少发表，因此很少以任何严谨的方式进行讨论。一位通过发表自己的观点树立良好榜样的批评家是约瑟夫·魏泽鲍姆，他的作品是《计算机能力与人类理性：从判断到计算》（旧金山：WH Freeman，1976 年）。
2. The critics and skeptics referred to here are distillations from years of public and private debates. These attitudes are widely held but, unfortunately, seldom published and therefore seldom discussed with any semblance of rigor. One critic who has set a good example by publishing his views is Joseph Weizenbaum in Computer Power and Human Reason: From Judgment to Calculations (San Francisco: W. H. Freeman, 1976).
不幸的是，魏泽鲍姆的书讨论了两个独立（但相关）的问题：计算机是否会损害人类的思维方式，以及计算机本身是否可以思考。魏泽鲍姆的大部分批评都集中在后一个问题上，他与休伯特·L·德雷福斯（Hubert L. Dreyfus）在《计算机不能做什么：对人工智能的批判》（纽约：Harper & Row，1972 年）中的观点一致。
Unfortunately Weizenbaum’s book discusses two separate (though related) questions: whether computers harm the way people think and whether computers themselves can think. Most critical reviews of Weizenbaum have focused on the latter question, on which he joins company with Hubert L. Dreyfus, What Computers Can’t Do: A Critique of Artificial Reason (New York: Harper & Row, 1972).
帕梅拉·麦考达克 (Pamela McCorduck) 在《会思考的机器》(Machines Who Think)（旧金山：WH Freeman，1979 年）一书中对关于计算机能否思考的争论中的一些主要参与者进行了生动的描述。
A lively description of some of the principal participants in the debate about whether computers can or cannot think is found in Pamela McCorduck, Machines Who Think (San Francisco: W. H. Freeman, 1979).
关于计算机是否真的影响人们的思维方式，目前发表的数据很少。S. Turkle 目前正在研究这个问题。
There is little published data on whether computers actually affect how people think. This question is being studied presently by S. Turkle.
3.许多版本的BASIC都允许程序生成类似LOGO程序HOUSE所创建的形状。最简单的示例如下所示：
3. Many versions of BASIC would allow a program to produce a shape like that made by the LOGO program HOUSE. The simplest example would look something like this:
10地块(0,0)
10 PLOT (0,0)
20地块(100,0)
20 PLOT (100,0)
30地块(100,100)
30 PLOT (100,100)
40地块(75,150)
40 PLOT (75,150)
50地块(0,100)
50 PLOT (0,100)
60地块(0,0)
60 PLOT (0,0)
70结束
70 END
编写这样的程序在许多方面都不如LOGO程序作为初级编程经验。它对初学者的要求更高，特别是需要笛卡尔坐标方面的知识。如果编写的程序可以成为其他项目的强大工具，那么这种要求就不会那么严格。LOGO程序SQ、TRI 和 HOUSE 可用于在屏幕上以任何位置和方向绘制正方形、三角形和房屋。BASIC程序允许在一个位置绘制一栋特定的房子。为了编写一个可以在许多位置绘制房屋的BASIC程序，必须使用代数变量，如PLOT ( x, y )、PLOT ( x + 100, y ) 等等。至于定义新命令，如SQ、TRI和HOUSE，常用的BASIC版本要么完全不允许这样做，要么至多允许通过使用高级技术编程方法实现类似的功能。BASIC的支持者可能会这样回答：(1) 这些反对意见仅指初学者的经验；(2) BASIC的这些缺陷是可以修正的。第一个论点根本不正确： BASIC的智力和实践原始性一直延伸到最先进的编程。第二个论点没有抓住我抱怨的重点。当然，人们可以将BASIC变成LOGO或SMALLTALK或其他任何东西，但仍然称之为“ BASIC ”。我的抱怨是，强加给教育界的东西并没有那么“修正”。此外，这样做有点像“改造”一座木屋，使其成为摩天大楼。
Writing such a program falls short of the LOGO program as a beginning programming experience in many ways. It demands more of the beginner, in particular, it demands knowledge of cartesian coordinates. This demand would be less serious if the program, once written, could become a powerful tool for other projects. The LOGO programs SQ, TRI, and HOUSE can be used to draw squares, triangles, and houses in any position and orientation on the screen. The BASIC program allows one particular house to be drawn in one position. In order to make a BASIC program that will draw houses in many positions, it is necessary to use algebraic variables as in PLOT (x, y), PLOT (x + 100, y), and so on. As for defining new commands, such as SQ, TRI, and HOUSE, the commonly used versions of BASIC either do not allow this at all or, at best, allow something akin to it to be achieved through the use of advanced technical programming methods. Advocates of BASIC might reply that: (1) these objections refer only to a beginner’s experience and (2) these deficiencies of BASIC could be fixed. The first argument is simply not true: The intellectual and practical primitivity of BASIC extends all along the line up to the most advanced programming. The second misses the point of my complaint. Of course one could turn BASIC into LOGO or SMALLTALK or anything else and still call it “BASIC.” My complaint is that what is being foisted on the education world has not been so “fixed.” Moreover, doing so would be a little like “remodeling” a wooden house to become a skyscraper.
第二章
CHAPTER 2
1. “思想实验”在科学，尤其是物理学中发挥了重要作用。如果在教育思考中更多地运用这些实验，将会鼓励人们采取更多的批判态度。
1. “Gedanken experiments” have played an important role in science, particularly in physics. These experiments would encourage more critical attitudes if used more often in thinking about education.
2.这里有个笑话。不熟悉诺姆·乔姆斯基近期作品的读者可能不明白。诺姆·乔姆斯基认为我们有一种语言习得装置。我不这么认为：MAD似乎并不比LAD更不可能。请参阅 N. Chomsky 的《语言反思》（纽约：万神殿，1976 年），了解他对大脑由与特定智力功能相匹配的专门神经器官组成的看法。我认为，教育未来的根本问题不是大脑是“通用计算机”还是专用设备的集合，而是我们的智力功能是否可以一一归结为神经学给定的结构。
2. There is a joke here. Readers who are not familiar with Noam Chomsky’s recent work may not get it. Noam Chomsky believes that we have a language acquisition device. I don’t: the MAD seems no more improbable than the LAD. See N. Chomsky, Reflections on Language (New York: Pantheon, 1976) for his view of the brain as made up of specialized neurological organs matched to specific intellectual functions. I think that the fundamental question for the future of education is not whether the brain is “a general purpose computer” or a collection of specialized devices, but whether our intellectual functions are reducible in a one-to-one fashion to neurologically given structures.
大脑拥有众多与生俱来的“小玩意”，这一点似乎是毋庸置疑的。但这些“小玩意”肯定比LAD和MAD等名称所暗示的要原始得多。我认为学习语言或学习数学就是要利用众多“小玩意”来实现这一目的，而这些“小玩意”的最初目的与它们所服务的复杂智力功能毫无相似之处。
It seems to be beyond doubt that the brain has numerous inborn “gadgets.” But surely these “gadgets” are much more primitive than is suggested by names like LAD and MAD. I see learning language or learning mathematics as harnessing to this purpose numerous “gadgets” whose original purpose bears no resemblance to the complex intellectual functions they come to serve.
第三章
CHAPTER 3
1.由于本书是为那些不太懂数学的读者编写的，因此对具体数学的引用尽可能少。以下评论将为数学经验丰富的读者充实讨论。
1. Since this book is written for readers who may not know much mathematics, references to specific mathematics are as restrained as possible. The following remarks will flesh out the discussion for mathematically sophisticated readers.
不同 Turtle 系统的同构是 Turtle 几何中出现的众多“高级”数学思想的例子之一，这些思想既具体又实用。其中，“微积分”中的概念尤为重要。
The isomorphism of different Turtle systems is one of many examples of “advanced” mathematical ideas that come up in Turtle geometry in forms that are both concrete and useful. Among these, concepts from “calculus” are especially important.
示例 1：积分。海龟几何学为线积分的概念铺平了道路，因为海龟在移动过程中经常会出现需要对某个量进行积分的情况。孩子们遇到的第一个情况通常是让海龟跟踪它转了多少圈或移动了多远。一个优秀的海龟项目是模拟向性，这种向性会导致动物寻找诸如温暖、光线或营养浓度等条件，并以位置数值函数的形式表示为场。通过沿海龟的路径对场量进行积分，很自然地会想到比较两种算法。一个简单的版本是通过在程序中插入一行来实现的，例如：CALL (: TOTAL + FIELD ) “ TOTAL ”，这意味着：取先前称为“ TOTAL ”的量，将量FIELD添加到其中，并将结果称为“ TOTAL ”。如果海龟采取的步骤太大或大小可变，此版本会出现“错误”。通过在遇到此类问题时进行调试，学生可以朝着更复杂的积分概念迈进。
Example 1: Integration. Turtle geometry prepares the way for the concept of line integral by the frequent occurrence of situations where the Turtle has to integrate some quantity as it goes along. Often the first case encountered by children comes from the need to have the Turtle keep track of how much it has turned or of how far it has moved. An excellent Turtle project is simulating tropisms that would cause an animal to seek such conditions as warmth, or light, or nutrient concentration represented as a field in the form of a numerical function of position. It is natural to think of comparing two algorithms by integrating the field quantity along the Turtle’s path. A simple version is achieved by inserting into a program a single line such as: CALL (:TOTAL + FIELD) “TOTAL”, which means: take the quantity previously called “TOTAL,” add to it the quantity FIELD and call the result “TOTAL.” This version has a “bug” if the steps taken by the Turtle are too large or of variable size. By debugging when such problems are encountered, the student moves in a meaningful progression to a more sophisticated concept of integral.
早期引入的沿路径积分的简单版本表明，一种经常出现的现象就是看似“自然”的教学顺序被逆转。在传统课程中，线积分是一个高级主题，学生在学习之前，必须先被鼓励将定积分视为曲线下的面积，而这个概念在纸笔技术的数学世界中似乎更为具体。但其结果是形成了对积分的误导性印象，当许多学生遇到用曲线下面积表示的积分时，他们会感到不知所措，因为这对于积分来说并不合适。
The early introduction of a simple version of integration along a path illustrates a frequent phenomenon of reversal of what seemed to be “natural” pedagogic ordering. In the traditional curriculum, line integration is an advanced topic to which students come after having been encouraged for several years to think of the definite integral as the area under a curve, a concept that seemed to be more concrete in a mathematical world of pencil and paper technology. But the effect is to develop a misleading image of integration that leaves many students with a sense of being lost when they encounter integrals for which the representation as area under a curve is quite inappropriate.
例 2：微分方程。 “触摸传感器海龟”（见介绍，注释 4）使用了一种让许多孩子感到兴奋而强大的方法。编写一个海龟绕物体飞行的典型第一种方法是测量物体并将其尺寸构建到程序中。因此，如果物体是一个边长为 150 个海龟步长的正方形，程序将包含FORWARD 150 指令。即使它有效（通常无效），这种方法也缺乏通用性。前面注释中引用的程序通过采取仅取决于海龟附近条件的微小步骤来工作。它不使用“全局”操作FORWARD 150，而只使用“局部”操作，例如FORWARD 1。这样做抓住了微分方程概念的基本核心。我见过小学生清楚地理解为什么微分方程是运动定律的自然形式。这里我们看到了另一个戏剧性的教学逆转：微分方程的力量在微积分的解析形式之前就被理解了。关于海龟版数学思想的大部分已知知识都汇集在 H. Abelson 和 A. diSessa 的《海龟几何：计算作为探索数学的媒介》（剑桥：麻省理工学院出版社，1981 年）中。
Example 2: Differential Equation. “Touch Sensor Turtle” (See introduction, note 4) used a method that strikes many children as excitingly powerful. A typical first approach to programming a Turtle to circumnavigate an object is to measure the object and build its dimensions into the program. Thus if the object is a square with side 150 Turtle steps, the program will include the instruction FORWARD 150. Even if it works (which it usually does not) this approach lacks generality. The program cited in the earlier note works by taking tiny steps that depend only on conditions in the Turtle’s immediate vicinity. Instead of the “global” operation FORWARD 150 it uses only “local” operations such as FORWARD 1. In doing so it captures an essential core of the notion of differential equation. I have seen elementary schoolchildren who understand clearly why differential equations are the natural form of laws of motion. Here we see another dramatic pedagogic reversal: The power of the differential equation is understood before the analytic formalism of calculus. Much of what is known about Turtle versions of mathematical ideas is brought together in H. Abelson and A. diSessa, Turtle Geometry: Computation as a Medium for Exploring Mathematics (Cambridge: MIT Press, 1981).
示例 3：拓扑不变量。让一只乌龟绕着一个物体爬行，在爬行过程中“总计”它的转弯次数：右转算作正转，左转算作负转。无论物体的形状如何，结果都是 360°。我们将看到这个“全乌龟行程定理”既有用又精彩。
Example 3: Topological Invariant. Let a Turtle crawl around an object “totalizing” its turns as it goes: right turns counting as positive, left turns as negative. The result will be 360° whatever the shape of the object. We shall see that this Total Turtle Trip Theorem is useful as well as wonderful.
2.弗洛伊德使用了“自我协调”这一短语。它是“用来描述自我可以接受的本能或想法的术语：即与自我的完整性及其要求相兼容的本能或想法。”（参见 J. Laplanche 和 J–B. Pontalis 的《精神分析语言》（纽约：诺顿出版社，1973 年））
2. The phrase “ego-syntonic” is used by Freud. It is a “term used to describe instincts or ideas that are acceptable to the ego: i.e., compatible with the ego’s integrity and with its demands.” (See J. Laplanche and J–B. Pontalis, The Language of Psycho-analysis (New York: Norton, 1973.)
3. G. Polya， 《如何解决它》（纽约州花园城：Doubleday-Anchor，1954 年）；《数学中的归纳和类比》（新泽西州普林斯顿：普林斯顿大学出版社，1954 年）；以及《合理推理的模式》（新泽西州普林斯顿：普林斯顿，1969 年）。
3. G. Polya, How to Solve It (Garden City, N.Y.: Doubleday-Anchor, 1954); Induction and Analogy in Mathematics (Princeton, N.J.: Princeton University Press, 1954); and Patterns of Plausible Inference (Princeton, N.J.: Princeton, 1969).
4.曲率的通常定义看起来更复杂，但与此定义等同。因此，我们有另一个以可理解形式呈现的“高级”概念的例子。
4. Usual definitions of curvature look more complex but are equivalent to this one. Thus we have another example of an “advanced” concept in graspable form.
5.如果可以向左或向右转弯，则必须将其中一个方向视为负方向。“（连通）区域的边界”是“简单闭合曲线”的简单说法。如果取消限制，转弯次数之和仍必须是 360 的整数倍。
5. If turns can be right or left, one direction must be treated as negative. “Boundary of (connected) area” is a simple way of saying “simple closed curve.” If the restriction is lifted, the sum of turns must still be an integral multiple of 360.
第四章
CHAPTER 4
1.在这里，我与杰里·布鲁纳发生了一点小争执。但我同意他的大部分想法，这不仅适用于语言和行动，也适用于文化材料与教学的学习关系。
1. Here I am picking a little quarrel with Jerry Bruner. But I share much of what he thinks, and this is true not only about language and action but also about the relationship to learning of cultural materials and of teaching.
通过比较我们对数学教育的态度，我们之间的系统性差异最为明显。布鲁纳作为一名心理学家，将数学视为一个既定实体，并以他独特的丰富方式考虑教授和学习数学的过程。我试图创造一种可学习的数学。我认为，在语言和文化方面，类似的东西将我们区分开来，并导致我们走向“学习理论”的不同范式。参见 JS Bruner 的《教学理论》 （剑桥：哈佛大学出版社，1966 年）和 JS Bruner 等人的《认知成长研究》（纽约：约翰·威利，1966 年）。
The systematic difference betweeen us is seen most clearly by comparing our approaches to mathematics education. Bruner, as a psychologist, takes mathematics as a given entity and considers, in his particular rich way, the processes of teaching it and learning it. I try to make a learnable mathematics. I think that something of the same sort separates us in regard to language and to culture and leads us to different paradigms for a “theory of learning.” See J. S. Bruner, Toward a Theory of Instruction (Cambridge: Harvard University Press, 1966) and J. S. Bruner et al., Studies in Cognitive Growth (New York: John Wiley, 1966).
2.最系统的研究是H. Austin，“A Computational Theory of Physical Skill” (Ph.D. thesis, MIT, 1976)。
2. The most systematic study is in H. Austin, “A Computational Theory of Physical Skill” (Ph.D. thesis, MIT, 1976).
3.这些程序进一步扩展了我们对编程的认识。它们能够同时“并行”运行。没有包括这种扩展的编程形象与现代计算世界完全脱节。而一个局限于串行编程的孩子就被剥夺了实践和概念能力的来源。一旦孩子试图将动作引入程序，就会感受到这种剥夺。
3. These procedures introduce a further expansion in our image of programming. They are capable of running simultaneously, “in parallel.” An image of programming that fails to include this expansion is quite out of touch with the modern world of computation. And a child who is restricted to serial programming is deprived of a source of practical and of conceptual power. This deprivation is felt as soon as the child tries to introduce motion into a program.
例如，假设一个孩子希望在计算机屏幕上制作一部包含三个独立移动物体的电影。实现此目的的“自然”方法是为每个物体创建一个单独的程序并让这三个物体运行。“串行”计算机系统强制采用一种不太合乎逻辑的方式来实现此目的。通常，每个物体的运动将被分解为几个步骤，并创建一个程序以循环顺序运行每个运动的步骤。
Suppose, for example, that a child wishes to create a movie on the computer screen with three separate moving objects. The “natural” way to do this would be to create a separate procedure for each object and set the three going. “Serial” computer systems force a less logical way to do this. Typically, the motions of each object would be broken up into steps and a procedure created to run a step of each motion in cyclic order.
这个例子说明了为什么儿童计算机系统应该允许并行计算或“多处理”的两个原因。首先，从工具的角度来看，多处理使复杂系统的编程更容易，概念更清晰。串行编程打破了本应具有自身完整性的程序实体。其次，作为一种学习模型，串行编程做得更糟：它背叛了模块化原则，阻碍了真正结构化的编程。儿童应该能够单独构建每个动作，尝试它，调试它，并知道它将作为更大系统的一部分工作（或几乎工作）。
The example shows two reasons why a computer system for children should allow parallel computation or “multi-processing.” First, from an instrumental point of view, multi-processing makes programming complex systems easier and conceptually clearer. Serial programming breaks up procedural entities that ought to have their own integrity. Second, as a model of learning, serial programming does something worse: It betrays the principle of modularity and precludes truly structured programming. The child ought to be able to construct each motion separately, try it out, debug it, and know that it will work (or almost work) as a part of the larger system.
多处理比简单的串行处理对计算资源的要求更高。学校和家庭中常见的计算机都没有足够的能力来实现多处理。早期的LOGO系统是“纯串行”的。较新的系统允许受限制的多处理形式（例如本章后面描述的WHEN DEMONS ），专门用于编程动态图形、游戏和音乐。在撰写本书时，开发一种限制更少的儿童多处理语言是MIT LOGO小组的主要研究目标。在这项工作中，我们大量借鉴了 Alan Kay 的SMALLTALK语言、Carl Hewitt 的“ ACTOR ”语言概念以及 Minsky-Papert 的“社会心理理论”中发展起来的思想。但是，此类系统固有的技术问题尚未完全理解，在就实现真正适合儿童的多处理系统的正确方法（或方法集）达成共识之前，可能需要进行更多的研究。
Multi-processing is more demanding of computational resources than simple serial processing. None of the computers commonly found in schools and homes are powerful enough to allow it. Early LOGO systems were “purely serial.” More recent ones allow restricted forms of multi-processing (such as the WHEN DEMONS described later in this chapter) tailored for purposes of programming dynamic graphics, games, and music. The development of a much less restrictive multi-processing language for children is a major research goal of the MIT LOGO Group at the time of writing this book. In the work we draw heavily on ideas that have been developed in Alan Kay’s SMALLTALK language, on Carl Hewitt’s concepts of “ACTOR” languages, and on the Minsky-Papert “Society Theory of Mind.” But the technical problems inherent in such systems are not fully understood and much more research may be needed before a concensus emerges about the right way (or set of ways) to achieve a really good multi-processing system suitable for children.
第五章
CHAPTER 5
1.对此类系统开发贡献最大的是 Andrea diSessa，除其他外，他还对“Dynaturtle”一词做出了贡献。H. Abelson 和 A. diSessa，《海龟几何：计算作为探索数学的媒介》（剑桥：麻省理工学院出版社，1981 年）。
1. The most prolific contributor to the development of such systems is Andrea diSessa, who is responsible, among many other things, for the term “Dynaturtle.” H. Abelson and A. diSessa, Turtle Geometry: Computation as a Medium for Exploring Mathematics (Cambridge: MIT Press, 1981).
2.猴子问题的讨论使用了一种计算模型。然而，这种模型与大多数编程语言中内置的算法编程计算概念相去甚远。建立这种模型包括创建对象集合并在它们之间建立交互。这种计算图像，后来被称为“面向对象”或“消息传递”编程，最初是作为模拟程序的技术方法开发的，并作为一种名为SIMULA的语言实现。最近它引起了更广泛的关注，尤其是成为人工智能研究的关注焦点，其中 Carl Hewitt 和他的学生对其进行了最广泛的开发。长期以来，Alan Kay 一直是教育领域面向对象语言的最积极倡导者。
2. The discussion of the Monkey Problem uses a computational model. However, this model is very far from fitting the notion of computation as algorithmic programming built into most programming languages. Making this model consists of creating a collection of objects and setting up interactions between them. This image of computation, which has come to be known as “object-oriented” or “message-passing” programming, was first developed as a technical method for simulation programs and implemented as a language called SIMULA. Recently it has drawn much broader interest and, in particular, has become a focus of attention in artificial intelligence research where it has been most extensively developed by Carl Hewitt and his students. Alan Kay has for a long time been the most active advocate of object-oriented languages in education.
第六章
CHAPTER 6
1.马丁·加德纳，《数学嘉年华》（纽约：兰登书屋，1977年）。
1. Martin Gardner, Mathematical Carnival (New York: Random House, 1977).
第七章
CHAPTER 7
1.皮亚杰 (Piaget) 对布尔巴基 (Bourbaki) 的评论，请参阅“Logique et connaissance scientifique”，编辑。 J. 皮亚杰 (J. Piaget)，普莱德百科全书，卷。 22（巴黎：伽利玛，1967 年）。
1. For remarks by Piaget on Bourbaki see “Logique et connaissance scientifique,” ed. J. Piaget, Encyclopidie de la Pleide, vol. 22 (Paris: Gallimard, 1967).
2. C. Lévi-Strauss，《结构人类学》，两卷本（纽约：Basic Books，1963-76 年）。
2. C. Lévi-Strauss, Structural Anthropology, 2 vols. (New York: Basic Books, 1963–76).
3.列维-斯特劳斯使用“拼凑”一词作为我们一直在讨论的类似修补过程的专业术语。“拼凑者”是指从事拼凑的人。这些概念是在罗伯特·劳勒的《一个孩子的学习：一项亲密研究》（博士论文，麻省理工学院，1979 年）中以计算背景发展起来的。
3. Lévi-Strauss uses the word bricolage as a technical term for the tinkering-like process we have been discussing. Bricoleur is the word for someone who engages in bricolage. These concepts have been developed in a computational context in Robert Lawler, “One Child’s Learning: An Intimate Study” (Ph.D. thesis, MIT, 1979).
4.当然，我们的文化为每个人提供了大量的机会来练习特定的系统程序。它缺乏思考和谈论程序的材料。当孩子们接触LOGO时，他们常常难以将程序识别为实体。在我看来，这样做类似于婴儿时期形成永久物体的过程，以及所有皮亚杰所保留的实体，如数字、重量和长度。在LOGO中，程序是可操作的实体。它们可以被命名、存储、检索、更改，用作超级程序的构建块，并分析为子程序。在这个过程中，它们被同化为更熟悉的实体的示意图或框架。因此，它们获得了“作为实体”的品质。它们继承了“具体性”。它们还继承了特定的知识。
4. Of course our culture provides everyone with plenty of occasions to practice particular systematic procedures. Its poverty is in materials for thinking about and talking about procedures. When children come to LOGO they often have trouble recognizing a procedure as an entity. Coming to do so is, in my view, analogous to the process of formation of permanent objects in infancy and of all the Piagetionly-conserved entities such as number, weight, and length. In LOGO, procedures are manipulable entities. They can be named, stored away, retrieved, changed, used as building blocks for superprocedures, and analyzed into subprocedures. In this process they are assimilated to schematic or frames of more familiar entities. Thus they acquire the quality of “being entities.” They inherit “concreteness.” They also inherit specific knowledge.